quantum critical response: from conformal perturbation theory to … · 2017. 5. 2. · abstract:...
TRANSCRIPT
Prepared for submission to JHEP
Quantum critical response: from conformal
perturbation theory to holography
Andrew Lucas,a Todd Sierensb,c and William Witczak-Krempad
aDepartment of Physics, Stanford University, Stanford, CA 94305, USAbPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, CanadacDepartment of Physics & Astronomy and Guelph-Waterloo Physics Institute, University of
Waterloo, Waterloo, Ontario N2L 3G1, CanadadDepartment of Physics and Regroupement quebequois sur les materiaux de pointe, Universite
de Montreal, Montreal, Quebec, H3C 3J7, Canada
E-mail: [email protected], [email protected],
Abstract: We discuss dynamical response functions near quantum critical points,
allowing for both a finite temperature and detuning by a relevant operator. When the
quantum critical point is described by a conformal field theory (CFT), conformal per-
turbation theory and the operator product expansion can be used to fix the first few
leading terms at high frequencies. Knowledge of the high frequency response allows
us then to derive non-perturbative sum rules. We show, via explicit computations,
how holography recovers the general results of conformal field theory, and the asso-
ciated sum rules, for any holographic field theory with a conformal UV completion –
regardless of any possible new ordering and/or scaling physics in the IR. We numer-
ically obtain holographic response functions at all frequencies, allowing us to probe
the breakdown of the asymptotic high-frequency regime. Finally, we show that high
frequency response functions in holographic Lifshitz theories are quite similar to their
conformal counterparts, even though they are not strongly constrained by symmetry.
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Contents
1 Introduction 1
1.1 Main Results 3
1.2 Outline 5
2 Conformal Perturbation Theory 6
2.1 Scalar Two-Point Functions 6
2.2 Conductivity 10
2.3 Viscosity 11
3 Holography: Bulk Action 12
3.1 Fixing CFT Data 15
3.1.1 COO 15
3.1.2 AO0O0O 16
3.1.3 AJJO 17
3.1.4 ATTO 19
4 Holography: High Frequency Asymptotics 19
4.1 Scalar Two-Point Functions 19
4.1.1 Subleading Orders in the Expansion 22
4.2 Conductivity 23
4.3 Viscosity 25
5 Holography: Full Frequency Response 27
5.1 A Minimal Model 27
5.2 Scalar Two-Point Functions 30
5.3 Conductivity 30
5.3.1 Reissner-Nordstrom Model 33
6 Lifshitz Holography 37
6.1 Bulk Model 38
6.2 Asymptotics of the Conductivity 39
6.3 Three-Point Function and (Holographic) Lifshitz Perturbation Theory 40
7 Sum Rules 42
8 Conclusion 44
– i –
A Two-Point Functions when ∆−D/2 is an Integer 46
A.1 ∆ = D/2 46
A.1.1 Conformal Perturbation Theory 46
A.1.2 Holography 47
A.2 ∆ > D/2 49
B Evaluating CabcdCabcd 49
C Computing ATTO 50
D Generalization of Holographic Asymptotic Integrals 52
D.1 a+ 2 < 2b 52
D.2 b < 0 53
E Viscosity in a Holographic Lifshitz Model 53
1 Introduction
A quantum critical point (QCP) is a zero-temperature continuous phase transition
which arises as a coupling parameter is tuned through a critical value. Typically, one
finds that at zero temperature (away from the QCP) there are two distinct ‘conven-
tional’ gapped or gapless phases of matter, whereas at finite temperature there is a finite
region of the phase diagram dominated strongly by critical fluctuations: see Figure 1.
A canonical example is the transverse field quantum Ising model in D = 1 + 1, 2 + 1
and 3+1 spacetime dimensions, where the QCP separates a paramagnet from a broken
symmetry phase, the ferromagnet. The QCP is described at low energies by the D-
dimensional Ising CFT [1]. Other examples involve fermions such as the Gross-Neveu
model [1, 2], and/or gauge fields (e.g. Banks-Zaks fixed points [3]).
Proximity to a QCP is believed to be the origin of exotic response functions in many
experimentally interesting condensed matter systems [4]. Even when the ground state
is conventional, so long as it lies ‘near’ a QCP, both thermodynamic and dynamical
response functions are significantly modified by quantum critical fluctuations. Because
interesting quantum critical theories are strongly coupled and lack long-lived quasi-
particle excitations, obtaining quantitative predictions for critical dynamical response
functions, where both quantum and thermal fluctuations must be accounted for, has
proven to be challenging. In this respect, quantum Monte Carlo studies have proven
useful to obtain the response functions in imaginary time [5–9]. The precise analytic
– 1 –
T
QCP
0 h
phase A phase B
T |h| 1D
quantum critical fan
Figure 1: Schematic phase diagram of a theory with a (conformal) quantum critical
point separating two phases at zero temperature. The non-thermal detuning parameter
h must be set to zero at the critical point. At finite temperature T , when T & |h| 1D−∆ =
|h|ν , there is a window in between the two phases where the system behaves qualitatively
like the quantum critical point h = 0, detuned only by temperature. In this paper,
we demonstrate that certain fingerprints in the dynamical response functions of the
quantum critical point can extend throughout this entire phase diagram.
continuation to real frequencies of such data constitutes a difficult open problem. The
results of this paper impose constraints on the response functions that will constrain
the continuation procedure.
In the case of conformal field theories (CFTs), one can derive rigorous and gen-
eral constraints on the dynamics at zero and finite temperature by using the operator
product expansion (OPE). In addition, one can study the response functions in the
neighborhood of the QCP by using conformal perturbation theory, where the expansion
parameter is the detuning of the coupling parameter from its critical value. Another
technique is the gauge-gravity duality [10], which relates certain large-N matrix quan-
tum field theories to classical gravity theories in higher dimensional spacetimes. The
power of this approach is that we are very easily able to access finite temperature and
finite frequency response functions, directly in real time. Unfortunately, experimentally
realized condensed matter systems do not have weakly coupled gravity duals, and some
care is required in identifying which features of these holographic theories are relevant
for broader field theories.
In this paper, we will discuss the asymptotics of a variety of response functions
at high frequency [5, 8, 11–20]. We will show that in holographic field theories, this
asymptotics is universal and does not depend in any detailed fashion on the nature of the
ground state. As a consequence, we will then derive non-perturbative sum rules which
– 2 –
may be tested directly in experiments. In the special case when the QCP is a conformal
field theory (CFT), we will also be able to compare our holographic calculation of the
response functions with an independent computation using techniques of conformal
field theory, and will find perfect agreement. We will also discuss the generalization
of these calculations to a richer class of Lifshitz field theories, which are less sharply
constrained by symmetry.
1.1 Main Results
We study a quantum critical system in D spacetime dimensions deformed by a relevant
operator O:
S = SQCP − h∫
dDx O, (1.1)
where for simplicity we have chosen the coupling h to be x-independent. We briefly
discuss the case of an inhomogeneous coupling in the conclusion 8. This formula is
presented in real time; in Euclidean time, there is a relative minus sign for the second
term. Now, let ∆ be the scaling dimension of O:
〈O(x, t)O(y, t)〉0 = COO|x− y|−2∆, (1.2)
where the 0 subscript means that the correlation function is evaluated at the QCP
T = h = 0. The parameter h is commonly called the (non-thermal) detuning parameter;
it drives the quantum phase transition. We also will study this detuned theory at
finite temperature T . The dynamical critical exponent z of this QCP is defined as
the ‘dimension’ of energy: namely, a dilation of space x → λx is accompanied by a
temporal dilation t→ λzt in order to be a symmetry. This implies that the excitations
of this theory (generally there are no well-defined quasi-particles at the QCP) disperse
as ω(q) ∼ qz. For simplicity, in much of this paper we will focus on the case z = 1,
specializing to the case where the QCP is a conformal field theory, returning to the
more general case z 6= 1 at the end of the paper.
Let X be a local operator of scaling dimension ∆X in this critical theory, and assume
for now that ∆X 6= (D/2)+n, where n is an integer, which is the generic case and ensures
that both powers in the brackets do not differ by an integer (otherwise, additional
logarithms can appear). We will show that, when z = 1, the generic asymptotic
expansion of the dynamical response function 〈XX〉 at high frequency is:
〈X (ω)X (−ω)〉 = (iω)2∆X−D[CXX +A h
(iω)D−∆+ B 〈O〉
(iω)∆+ · · ·
], (1.3)
where 〈· · · 〉 denotes an expectation value evaluated at non-zero values of h and T ,
in contrast to 〈· · · 〉0, which is evaluated directly in the quantum critical theory, at
– 3 –
CFT operator X A/BTxy (stress tensor) (2.41)
Jx (conserved current) (2.33)
O0 (scalar operator) (2.18)
Table 1: Where to find our main results for the ratio of coefficients A/B in the
asymptotic expansion of 〈XX〉 in CFTs, (1.3).
h = T = 0. We emphasize that the first correction is analytic in the coupling h
and independent of temperature. The expansion (1.3) holds when ω T, |h|1/(D−∆)
(~ = kB = 1) and when the detuned system has a finite correlation length. It can
fail in regions separated from the quantum critical “fan” by a phase transition, where
potentially new gapless modes can arise. An example of this failure was shown in the
broken symmetry (Goldstone) phase in the vicinity of the Wilson-Fisher O(N) QCP
at large N [20]. The leading order coefficient CXX is an operator normalization in
the vacuum of the CFT. We compute the coefficients A and B exactly for arbitrary
spacetime dimension D and dimension ∆ of the (relevant) operator, in three practical
examples: 〈O0O0〉, 〈JxJx〉, and 〈T xyT xy〉. Here O0 is a scalar operator, Jx is a spatial
component of a conserved current, and T xy is an off-diagonal spatial component of the
conserved stress-energy tensor. Remarkably, A and B will not depend on either h or
T , and are properties of the pure CFT, and the choice of operator X . Furthermore,
the ratio A/B is universal, depending only on ∆ and D.
When ∆X and/or ∆X −∆ is an integer or half-integer, depending on the dimension
D, logarithmic corrections to (1.3) appear. These special cases are important because
conserved currents have such dimensions, and we will discuss them later in this paper.
The reason thatO be a relevant operator is to ensure that the asymptotic expansion
(1.3) is well-behaved; if O is irrelevant, the second term in (1.3) dominates. This is
not surprising, as we are probing the UV physics (which is sensitive to the dynamics
of irrelevant operators).
One of the main purposes of the asymptotic expansion (1.3) is to verify when the
following kind of sum rule holds:
∞∫0
dω Re
(〈X (−ω)X (ω)〉iω
− CXX · (iω)2∆X−D)
= 0. (1.4)
Clearly, this sum rule is automatically satisfied at the critical point, but it can also be
satisfied off-criticality. When h = 0, we will see that such a sum rule will only hold if
2∆X −D < ∆. When h 6= 0, an additional inequality ∆ < 2(D−∆X ) is also required.
– 4 –
Eq. (1.3) is presented in real time. Our derivation of this result will occur in
Euclidean time. It is usually straightforward to analytically continue back to real time.
There can be subtleties for certain values of D and ∆, for certain operators X , and we
will clarify the restrictions later.
We will present two derivations of (1.3): one using conformal perturbation theory
applied to general CFTs (such an approached was succinctly presented for the con-
ductivity in [20]), one using holography. We will show that these approaches exactly
agree. A priori, as the conformal perturbation theory approach does not rely on the
large-N matrix limit of holography, it may seem as though the holographic derivation
is superfluous. However, we will argue that this is not the case: that conformal per-
turbation theory is completely controlled at finite h and finite T is not always clear,
especially if the ground state of the detuned theory is not a conventional gapped phase.
Our holographic computation reveals that, independent of the details of the ground
state, the asymptotics of the response functions are given by (1.3), with all coefficients
in the expansion determined completely by the UV CFT. We expect that – with some
important exceptions in broken symmetry phases [20] – this is a robust result of our
analysis, valid beyond the large-N limit.
For holographic models, we will also study the case z 6= 1, and find a simple
generalization of (1.3). While we will prove the universality of (1.3) even beyond
holography when z = 1, we cannot prove this universality for z 6= 1.
We also present the full frequency dependence of these response functions in mul-
tiple holographic models. In addition to verifying that the asymptotics (1.3) and the
associated sum rules such as (1.4) hold, we will be able to probe frequency scales past
which conformal perturbation theory fails, both at a finite temperature critical point,
and in systems strongly detuned off-criticality.
1.2 Outline
The remainder of the paper is organized as follows. In Section 2, we will describe
the asymptotics of response functions using conformal perturbation theory. In Section
3, we will describe a very general class of holographic theories with conformal UV
completions, and compute all necessary CFT data required to compare with Section
2. Section 4 contains a direct computation of high frequency holographic response
functions, in which exact agreement with conformal perturbation theory is found. We
numerically compute the full frequency response functions in Section 5. We generalize
our approach in Section 6 to holographic theories with ‘Lifshitz’ field theory duals,
where there are, as of yet, no other approaches like conformal field theory which can
be used. We discuss sum rules in Section 7. We conclude the paper by discussing the
– 5 –
possible breakdown of (1.3) in broken symmetry phases, and the generalization of (1.3)
to QCPs deformed by disorder.
2 Conformal Perturbation Theory
In this section, we present a computation of the high-frequency asymptotic behavior of
the three advertised two-point functions, 〈O0O0〉, 〈JxJx〉, 〈T xyT xy〉, using conformal
perturbation theory for generic spacetime dimension D. The computation of 〈JxJx〉was recently presented by two of us [20]. We will present the computation of 〈O0O0〉in detail – the computation of the latter two is extremely similar, and so we simply
summarize the results.
2.1 Scalar Two-Point Functions
We begin with the correlator 〈O0O0〉 of a scalar operator O0 with dimension ∆0. In
the vacuum of a unitary CFT, one finds that in position space
〈O0(x)O0(y)〉0 =CO0O0
|x− y|2∆0. (2.1)
In this work, we are interested in momentum space correlation functions, which are
readily computed (up to an overall δ function):
〈O0(k)O0(−k)〉0 =
∫dDx eik·x〈O0(x)O0(0)〉0 =
∞∫0
ds
s
∫dDxeik·x−sx2 s∆0
Γ(∆0)CO0O0
=2D−2∆πD/2Γ(D
2−∆0)
Γ(∆0)CO0O0k
2∆0−D ≡ CO0O0k2∆0−D. (2.2)
If ∆0 − D2
= 0, 1, 2, . . . is an integer, then we find an extra factor of log k in (2.2);
henceforth in the main text, we will assume that ∆0 − D2
is not an integer. We also
note that CO0O0 > 0, but depending on ∆ and D, CO0O0 can take either sign.
We can now deform away from the exact critical point by turning on a finite
temperature T or detuning h. (2.2) will now receive corrections due to h and T – the
leading order corrections are written in (1.3). We first outline the argument for how
the structure (1.3) arises, after which it will be relatively straightforward to do a direct
computation to fix A and B. Suppose that we set T = 0, for simplicity, but detune our
– 6 –
CFT by h, (1.1). We express the expectation value using a Euclidean path integral:
〈O0(Ω)O0(−Ω)〉h =1
Z[h]
∫DΦCFTe−SCFT+hO(0)O0(Ω)O0(−Ω)
=Z[0]
Z[h]
⟨O0(Ω)O0(−Ω)ehO(0)
⟩0
= 〈O0(Ω)O0(−Ω)(1 + hO(0) + · · · )〉0= CO0O0Ω2∆0−D +
AhΩD−∆
Ω2∆0−D + · · · (2.3)
Here O(0) is the zero momentum mode of O. The first term of (2.3) follows from (2.2).
The power of Ω in the second term has been fixed by dimensional analysis; below we
explicitly evaluate this term by evaluating the 3-point function 〈O0O0O〉0. This 3-point
function is fixed by the conformal Ward identities in momentum space up to an overall
coefficient; alternatively, one can perform an analytic continuation from position to
momentum space, although one must be careful with regularization [21]. Holography
naturally produces the correct answer as we show below. There are further terms in
(2.3) containing subleading integer powers of h/ΩD−∆.
However, the O0 2-point correlation function will also contain terms non-analytic
in h/ΩD−∆. It will often be the case, in the pure CFT, that the operator product
expansion (OPE) will take the form, here expressed in momentum-space:
O0(Ω)O0(−Ω) = CO0O0Ω2∆0−D + BΩ2∆0−DO(0)
Ω∆+ · · · (2.4)
We have 〈O〉0 = 0, since the pure CFT has no dimensionful parameter. Hence, upon
averaging this equation in a CFT, the B contribution to the OPE vanishes. However,
once we detune with h at T = 0, we expect:
〈O〉h = A±|h|∆
D−∆ , (2.5)
with a coefficient that generally depends on the sign of h. The OPE (2.4) is a high
energy property of the theory, and should be valid even at finite h, so long the frequency
Ω is large enough: h ΩD−∆. Hence, we expect that we may apply the expectation
value to the local OPE and find an additional contribution to the two-point function,
linear in 〈O〉:
〈O0(Ω)O0(−Ω)〉h = CO0O0Ω2∆0−D +Ah
ΩD−∆Ω2∆0−D + · · ·+ B〈O〉
Ω∆Ω2∆0−D + · · · . (2.6)
In the context of classical critical phenomena, an analogous expansion for the short-
distance spatial correlators of the order parameter have been found for thermal Wilson-
Fisher fixed points in D = 3 [22], and subsequently more quantitatively in other theories
– 7 –
using the OPE [23–25]. These classical results are a special case of (2.6) analytically
continued to imaginary times at T = 0. In the context of the classical 3D Ising model,
such an expansion was recently used together with conformal bootstrap results to make
predictions for the correlators of various scalars near the critical temperature Tc [25].
We also see that the coefficient B is nothing more than an OPE coefficient in the CFT,
as found in [8, 20].
Obviously, for generic ∆, the power of h in (2.5) is not an integer. Hence, the B term
in this expansion cannot be captured at any finite order in the conformal perturbation
theory expansion (2.3). Still, we will see that it is possible to compute the ratio A/Bin terms of CFT data.
We claim that upon generalizing to finite temperature T , the essential arguments
above follow through, despite the fact that when T > 0, 〈O〉 will be an analytic
function of h near h = 0. Our holographic computation will confirm this insight in a
broad variety of models.
The argument above contains essentially all of the physics of conformal perturba-
tion theory. It remains to explicitly fix the coefficient A in terms of B. To do so, we
carefully study the 3-point function 〈O0(p1)O0(p2)O(p3)〉0, upon choosing the momenta
to be
p1 = (Ω, 0), (2.7a)
p2 = (−Ω− p, 0), (2.7b)
p3 = (p, 0). (2.7c)
where p Ω. In a conformal field theory, this correlation function is completely fixed
up to an overall prefactor AO0O0O [21, 26]:
〈O0(p1)O0(p2)O(p3)〉0 = AO0O0OI(D2− 1,∆0 − D
2,∆0 − D
2,∆− D
2
)(2.8)
where we have defined (for later convenience):
I(a, b, c, d) ≡∞∫
0
dx xapb1pc2pd3 Kb(p1x)Kc(p2x)Kd(p3x). (2.9)
Here Kν(x) is the modified Bessel function which exponentially decays as x → ∞.
From (2.7), we see that p1,2 p3. Hence, a good approximation to (2.9) should come
from an asymptotic expansion of Kd(p3x) in (2.9). Defining
Z(b) ≡ 2b−1Γ(b) (2.10)
– 8 –
and using the asymptotic expansion
Kb(x) = Z(b)x−b[1 + O
(x2)]
+ Z(−b)xb[1 + O
(x2)], (2.11)
we take the limit of (2.7) and simplify (2.9) to
〈O0(p1)O0(p2)O(p3)〉0 ≈ AO0O0OZ(∆− D
2
)Ψ(D −∆; ∆0 − D
2
)Ω2∆0+∆−2D
+ AO0O0OZ(D2−∆
)Ψ(∆; ∆0 − D
2
)p2∆−DΩ2∆0−∆−D, (2.12)
where we have defined
Ψ(a; b) ≡∞∫
0
dx xa−1Kb(x)2 =
√πΓ(a
2)Γ(a
2+ b)Γ(a
2− b)
4Γ(1+a2
). (2.13)
The first term of (2.12) is completely regular as p → 0. The correlation function
〈O0(Ω)O0(−Ω)O(0)〉0 which appeared in conformal perturbation theory in (2.3) should
be interpreted as this first contribution. Thus we obtain
A = AO0O0OZ(∆− D
2
)Ψ(D −∆; ∆0 − D
2
). (2.14)
The second term of (2.12) can be interpreted as follows. At a small momentum p, the
OPE (2.4) generalizes to
O0(Ω)O0(−Ω− p) = BΩ2∆0−DO(−p)Ω∆
+ · · · . (2.15)
Upon contracting both sides of (2.15) with 〈· · · O(p)〉CFT, we obtain a contribution
which is non-analytic in p:
〈O0(p1)O0(p2)O(p3)〉0 = · · ·+ BΩ2∆0−∆−D〈O(−p)O(p)〉0= · · ·+ BC0p
2∆−DΩ2∆0−∆−D. (2.16)
Comparing (2.16) with (2.12) we conclude
BCOO = AO0O0OZ(D2−∆
)Ψ(∆; ∆0 − D
2
). (2.17)
The ratio A/B is independent of AO0O0O:
AB = COO
Z(∆− D
2
)Ψ(D −∆; ∆0 − D
2
)Z(D2−∆
)Ψ(∆; ∆0 − D
2
) . (2.18)
As promised, we have fixed A and B completely in terms of the CFT data ∆0, ∆, COOand AO0O0O.
The assumption that ∆ − D/2 is not an integer is relaxed in Appendix A. For
the most part, (1.3) is qualitatively unchanged, with an interesting exception when
∆ = D/2.
– 9 –
2.2 Conductivity
The conductivity at finite (Euclidean) frequency is defined as
σ(iΩ) ≡ − 1
Ω〈Jx(−Ω)Jx(Ω)〉, (2.19)
where the spatial momentum has been set to zero. Note that the operator dimension
∆J = D − 1 is fixed by symmetry. In a pure CFT, at zero temperature, we find (in
position space) [27]
〈Jµ(x)Jν(0)〉0 =CJJx2D−2
Iµν(x) (2.20)
with CJJ > 0 and
Iµν ≡ δµν − 2xµxνx2
. (2.21)
This can be Fourier transformed, analogous to (2.2), to give
〈Jµ(k)Jν(−k)〉0 = −CJJkD−2πµν(k) (2.22)
with
CJJ ≡ −22−DπD/2(D − 2)Γ(1− D
2)
Γ(D)CJJ (2.23)
and the transverse (Euclidean) projector
πµν(k) ≡ δµν −kµkνk2
. (2.24)
In this paper, we will generically take µ = ν = x and k0 = Ω, k = 0. Hence we find
σ(iΩ) = CJJΩD−3 (2.25)
in a pure CFT.
Actually, this argument was a bit fast. In even dimensions D = 4, 6, 8, . . ., the
coefficient CJJ diverges due to a pole in the Γ function. Upon regularization, this
implies a logarithm in the momentum space correlator, and so the conductivity:
σ(iΩ) = (−1)D2
+1CJJΩD−3 logΩ
Λ, (2.26)
with Λ a UV cutoff scale and
CJJ ≡ CJJ23−Dπ
D2 (D − 2)
Γ(D)Γ(D2
). (2.27)
– 10 –
Note that CJJ > 0, and so in real time (setting iΩ = ω) we find that the dissipative
real part of the conductivity is finite and independent of Λ:
Re (σ(ω)) =π
2CJJωD−3 > 0. (2.28)
The dimension D = 2 is special. Indeed, one finds that the limit D → 2 is regular in
(2.23): and hence in Euclidean time (at zero temperature)
σ(iΩ) =2πCJJ
Ω. (2.29)
Now, we are ready to perturb the pure CFT by a finite T and h. Upon doing
so, the asymptotics of σ(iΩ) at large Ω was recently computed in [20] using conformal
perturbation theory. Conformal invariance demands [21]:
〈Jx(p1)Jx(p2)O(p3)〉0 = AJJO[I(D2, D
2− 1, D
2− 1,∆− D
2+ 1)
+∆
2(D − 2−∆)I
(D2− 1, D
2− 1, D
2− 1,∆− D
2
)](2.30)
so long as the x-component of all momenta vanishes, as in (2.7). Following the exact
same procedure as the previous section, we find that (1.3) becomes
σ(iΩ) = ΩD−3
[CJJ +A h
ΩD−∆+ B〈O〉
Ω∆+ · · ·
](2.31)
with
A = −AJJO(D −∆)(1− ∆
2
)Z(∆− D
2
)Ψ(D −∆; D
2− 1), (2.32a)
B = −AJJOCOO∆(1− D−∆
2
)Z(D2−∆
)Ψ(∆; D
2− 1). (2.32b)
The ratioAB = COO
(D −∆)(2−∆)
∆(2 + ∆−D)
Z(∆− D2
)Ψ(D −∆; D2− 1)
Z(D2−∆)Ψ(∆; D
2− 1)
. (2.33)
2.3 Viscosity
The viscosity at finite (Euclidean) frequency is defined as
η(iΩ) ≡ − 1
Ω〈Txy(−Ω)Txy(Ω)〉. (2.34)
As with Jµ, the stress tensor Tµν has a protected operator dimension ∆T = D. In a
pure CFT, one finds in position space [27]
〈Tµν(x)Tρσ(0)〉0 =CTTx2D
(Iµρ(x)Iνσ(x) + Iµσ(x)Iνρ(x)
2− δµνδρσ
D
). (2.35)
– 11 –
After a Fourier transform to momentum space:
〈Tµν(k)Tρσ(−k)〉0 = −CTTkD(πµρ(k)πνσ(k) + πνρ(k)πµσ(k)
2− πµν(k)πρσ(k)
D − 1
)(2.36)
with
CTT = −D − 1
D + 1
πD2 Γ(−D
2)
2DΓ(D)CTT . (2.37)
We now find that in all even dimensions D, there are logarithmic corrections to this
correlation function.
At finite T and h, we follow the same arguments as above to determine A and B.
Conformal invariance demands [21]:
〈Txy(p1)Txy(p2)O(p3)〉0 = ATTO
[I
(D2
+ 1, D2− 1, D
2− 1,∆− D
2+ 2
)+ ∆+2
2(D −∆− 2)I
(D2, D
2− 1, D
2− 1,∆− D
2+ 1)
+∆
8(D −∆)(∆ + 2)(D − 2−∆)I
(D2− 1, D
2, D
2,∆− D
2
)](2.38)
so long as the momenta are of the form (2.7). Following an identical procedure to
before we find the asymptotic expansion at large frequencies
η(iΩ) = ΩD−1
[CTT +A h
ΩD−∆+ B〈O〉
Ω∆+ · · ·
](2.39)
with
A = ATTO∆(D −∆)(D −∆ + 2)(∆− 2)
8Z(∆− D
2
)Ψ(D −∆; D
2
), (2.40a)
B =ATTOCOO
∆(D −∆)(∆ + 2)(D −∆− 2)
8Z(D2−∆
)Ψ(∆; D
2
). (2.40b)
The ratioAB = COO
(D −∆ + 2)(∆− 2)
(D −∆− 2)(2 + ∆)
Z(∆− D2
)Ψ(D −∆; D2
)
Z(D2−∆)Ψ(∆; D
2)
. (2.41)
3 Holography: Bulk Action
Now, we begin our holographic derivation of these results. The “minimal” holographic
model which we will study has the following action (in Euclidean time):
S =
∫dD+1x
√g
[R
2κ2+
1
2(∂Φ)2 + V (Φ) +
Y (Φ)
κ2CabcdC
abcd +Z(Φ)
4e2FabF
ab
+1
2(∂ψ)2 +
W (Φ)
2ψ2 + · · ·
]. (3.1)
– 12 –
bulk field CFT operator
gµν (metric) Tµν (stress tensor)
Aµ (gauge field) Jµ (conserved U(1) current)
Φ O (relevant scalar)
ψ O0 (probe scalar)
Table 2: The four bulk fields of (3.1), and their dual operators in the CFT of interest.
Our interest will be in the influence of O (and hence Φ) on the two-point correlators
of Tµν , Jµ and O0.
As usual, R is the Ricci scalar, Fab is the field strength of a bulk gauge field Aµ, and Cabcdis the Weyl curvature tensor (traceless part of the Riemann tensor). Bulk dimensions
are denoted with ab · · · . We additionally have two scalar fields Φ and ψ, dual to Oand O0 respectively. The purpose of each bulk field is to source correlation functions of
different operators in the boundary theory, as described in Table 2. The functions V ,
Y , Z and W are all functions of the bulk scalar field Φ, dual to the operator O whose
influence on the asymptotic behavior of correlation functions is the focus of this paper.
We impose the following requirements:
V (Φ→ 0) = −D(D − 1)
2L2κ2+
∆(∆−D)
2L2Φ2 + · · · , (3.2a)
Z(Φ→ 0) = 1 + αZLD−1
2 Φ + · · · , (3.2b)
Y (Φ→ 0) = αYLD+3
2 Φ + · · · , (3.2c)
W (Φ→ 0) =∆0(∆0 −D)
L2+ αWL
D−52 Φ + · · · (3.2d)
Table 3 explains how the details of this model will be related to the CFT data of the
theory dual to (3.1). We will make these connections precise later in this section.
Unless otherwise stated, in this paper we will assume that the only non-trivial
background fields are the metric gµν and Φ. The field theory directions will be denoted
with xµ, and the emergent bulk radial direction with r. We take r → 0 to be the
asymptotically AdS region of the bulk spacetime, where an approximate solution to
the equations of motion from (3.1) is
ds2(r → 0) =L2
r2
(dr2 + dt2e + dx2
), (3.3)
and
Φ(r → 0) =1
LD−1
2
[hrD−∆ + · · ·+ 〈O〉
2∆−Dr∆ + · · ·
]. (3.4)
– 13 –
bulk coupling CFT data
LD−1/κ2 CTTLD−3/e2 CJJαZ CJJOαY CTTOαW CO0O0O
Table 3: The couplings in the holographic model (3.1) are related to the CFT data of
its dual theory. Here we have listed the qualitative connection between certain dimen-
sionless numbers associated with (3.1) and CFT data of interest for our computation.
One thing missing from this table is COO – this is related to the normalization of the
kinetic term for Φ in (3.1), which we have set to 12
as per usual.
With the normalizations in (3.4), h is exactly equal to the detuning parameter h as
defined in (1.1), and 〈O〉 is identically the expectation value of the dual operator
[10]. Exceptions arise when ∆−D/2 is an integer, as the holographic renormalization
procedure becomes more subtle [28]; these cases are discussed further in Appendix A.
Let us briefly comment on the logic behind the construction (3.1). The W , Z and
Y terms are chosen to be the simplest diffeomorphism/gauge invariant couplings of
Φ to ψ, Aa and gab respectively. In the last case, we further demand that Φ is not
sourced by the AdS vacuum – this forbids couplings such as R2, RabcdRabcd, etc. The
reason for this is that in a pure CFT, any relevant scalar operator O satisfies 〈O〉 = 0.
Hence, we cannot linearly couple Φ to any geometric invariant which is non-vanishing
on the background (3.3). Remarkably, we will see that the Y (Φ) coupling, which was
introduced in [19] so that 〈O〉 6= 0 at finite temperature, also is precisely the term in
the bulk action which leads to ATTO 6= 0.
We will be computing two point functions holographically. Hence, it is important
to understand the asymptotic behavior of ψ, Ax and gxy – related to 〈O0O0〉, σ and η
respectively. One finds
ψ =1
LD−1
2
[h0r
D−∆ + · · ·+ 〈O0〉2∆0 −D
r∆0 + · · ·], (3.5a)
Ax = A0x +
e2
LD−3〈Jx〉
rD−2
D − 2+ · · · , (3.5b)
gxy =L2
r2
[g0xy + · · ·+ 2κ2
LD−1〈Txy〉
rD
D+ · · ·
](3.5c)
where h0, A0x and g0
xy are the source operators for O0, Jx and Txy respectively. A0x
– 14 –
can be thought of as an external gauge field, and g0xy an external metric deformation
in the boundary theory. In practice, we will compute two-point functions by solving
appropriate linearized equations and looking at the ratio of response to source: for
example,
〈O0(−Ω)O0(Ω)〉 =〈O0〉h0
∣∣∣∣Ω
. (3.6)
Let us briefly note that for D = 2, the model we have introduced describes a
CFT with a marginal JµJµ deformation [29]. In this dimension, the expansion of
the gauge field in (3.5) becomes A0x + Le2〈Jx〉 log r; the coefficient A0
x is sensitive to
the ‘cutoff’ in the logarithm, and this ambiguity is related to logarithmically running
couplings associated with the marginal deformation. The asymptotic corrections to the
conductivity which we will compute in this paper are regular in the D → 2 limit.
3.1 Fixing CFT Data
Our ultimate goal is to compare a holographic computation of two-point functions to
the general expansion obtained using conformal field theory. First, however, we will
fix four CFT coefficients in the holographic model presented above: COO, AO0O0O,
AJJO and ATTO. As we are computing CFT data, we may assume in this section that
the background metric is given exactly by (3.3). Though we will compute two-point
correlators analogously to (3.6), we will compute three-point correlators by directly
evaluating the on-shell bulk action.
Although these are rather technical exercises, we will see a beautiful physical cor-
respondence between holographic Witten diagrams and the I(a, b, c, d) integrals over
Bessel functions, defined in (2.9) and found in [21] to be a consequence of conformal
invariance alone.
3.1.1 COOIn D spacetime dimensions in a Euclidean AdS background (3.3), the equation of
motion for the field Φ(r)eiΩt, dual to operator O(Ω), is:
0 =1√g∂r (√g∂rΦ)− Ω2gttΦ = m2Φ + · · · , (3.7)
which becomes
r2∂rΦ− (D − 1)r∂rΦ− Ω2r2Φ = ∆(∆−D)Φ + · · · . (3.8)
Further nonlinear terms in this equation of motion are not relevant for the present
computation. The regular solution in the bulk (which vanishes as r →∞) is given by
Φ = rD/2K∆−D/2(Ωr) = ArD−∆ +Br∆ + · · · . (3.9)
– 15 –
The two-point function 〈O(Ω)O(−Ω)〉 ≡ COOΩ2∆−D is proportional to the ratio B/A.
More precisely, using (2.11) and the definition of Z in (2.10), we find
〈O(−Ω)O(Ω)〉0 = (2∆−D)B
A= (2∆−D)Ω2∆−DZ(D
2−∆)
Z(∆− D2
)(3.10)
which fixes
COO = (2∆−D)Z(D
2−∆)
Z(∆− D2
). (3.11)
This derivation assumes that (as noted previously) ∆−D2
is not an integer; see Appendix
A for the changes to this derivation for this special case.
3.1.2 AO0O0O
It is easiest to compute three-point correlators in holography by directly evaluating the
classical bulk action on-shell (see e.g. [30, 31]):
〈O0(p1)O0(p2)O(p3)〉0 = − δ3Sbulk
δjO0(p1)δjO0(p2)δjO(p3)(3.12)
We are denoting here jO0 and jO as coefficients in the asymptotic expansion
Φ(p) =jO(p)rD−∆
LD−1
2
+ · · · , (3.13a)
ψ(p) =jO0(p)rD−∆0
LD−1
2
+ · · · . (3.13b)
To compute AO0O0O (to leading order in N), we know that the boundary-bulk prop-
agators for Φ and ψ are nothing more than the solutions to their free (in the bulk)
equations of motion. Hence, from (2.11) and (3.9):
Φ(p, r) =r
D2 p∆−D
2 K∆−D2
(pr)
LD−1
2 Z(∆− D2
)jO(p), (3.14a)
ψ(p, r) =r
D2 p∆0−D
2 K∆0−D2
(pr)
LD−1
2 Z(∆0 − D2
)jO0(p). (3.14b)
– 16 –
Hence,
−〈O0(p1)O0(p2)O(p3)〉0 =δ3
δjO0(p1)δjO0(p2)δjO(p3)
∫dD+1x
√gαW2L
D−52 ×[∫
dDp1eip1·xψ(p1, r)
] [∫dDp2eip2·xψ(p2, r)
] [∫dDp3eip3·xΦ(p3, r)
]=
∫drdDx
(L
r
)D+1
αWLD−5
2 ei(p1+p2+p3)·x×
rD2 p
∆0−D2
1 K∆0−D2
(p1r)
LD−1
2 Z(∆0 − D2
)
rD2 p
∆0−D2
2 K∆0−D2
(p2r)
LD−1
2 Z(∆0 − D2
)
rD2 p
∆−D2
3 K∆−D2
(p3r)
LD−1
2 Z(∆− D2
)
=αW δ
D(p1 + p2 + p3)
Z(∆0 − D2
)2Z(∆− D2
)I(D2− 1,∆0 − D
2,∆0 − D
2,∆− D
2
). (3.15)
Comparing with (2.8), we conclude that
AO0O0O = − αW
Z(∆0 − D2
)2Z(∆− D2
). (3.16)
3.1.3 AJJO
This computation proceeds similarly to before, but is a bit more technically involved.
In this subsection and in the next, we assume that all momenta are in the t direction,
which simplifies many tensorial manipulations. The Ax propagator in the bulk is
Ax(p, r) =KD
2−1(pr)
Z(D2− 1)
jAx(p), (3.17)
where jAx is the boundary source for the dual current operator Jx, and we have defined,
for later convenience,
Kb(x) = xbKb(x). (3.18)
(3.17) follows from the bulk Maxwell equations in pure AdS:
0 =1√g∂a (√gF ax) =
(L
r
)3−D
∂r
((L
r
)D−3
∂rAx
)− p2Ax. (3.19)
Now, we need to evaluate the ΦAxAx contribution to the bulk action, just as before.
This is
Sbulk =
∫dD+1x
√gαZL
D−12
e2Φ(p3)
[gabgxx∂aAx(p1)∂bAx(p2)
]. (3.20)
– 17 –
Now, suppose we integrate by parts to remove derivatives from Ax(p1):
Sbulk = −αZLD−1
2
e2
∫dD+1x
√gAx(p1)gxxgab∂bAx(p2)∂aΦ(p3)
− αZLD−1
2
e2
∫dD+1xAx(p1)Φ(p3)∂a
[√ggabgxx∂bAx(p2)
]. (3.21)
The second line of the above equation vanishes by (3.19). Now, we combine (3.21) with
an equivalent equation where we integrate by parts to remove derivatives from Ax(p2),
leading to:
Sbulk = −αZLD−1
2
2e2
∫dD+1x
√ggxxgab∂aΦ(p3)∂b(Ax(p1)Ax(p2)). (3.22)
We now integrate by parts to remove all derivatives from Ax:
Sbulk =αZL
D−12
2e2
∫dD+1xAx(p1)Ax(p2)gxx
[∂a(√
ggab∂bΦ(p3))
+2
r
√ggrr∂rΦ(p3)
]=αZL
D−12
2e2
∫dD+1x
√gAx(p1)Ax(p2)gxx
[∆(∆−D)
L2Φ(p3) +
2
rgrr∂rΦ(p3)
],
(3.23)
where we have employed (3.8) in the second step. Using the derivative identities
∂x(xbKb(x)
)= −xbKb−1(x) = 2bxb−1Kb(x)− xbKb+1(x), (3.24a)
∂x(xbIb(x)
)= xbIb−1(x), (3.24b)
and the explicit expression (3.13) for Φ(p3), we obtain
∂rΦ(p3) =rD−∆−1
LD−1
2 Z(∆− D2
)
[∆K∆−D
2(p3r)− K∆−D
2+1(p3r)
](3.25)
Hence, we find
Sbulk = −δD(p1 + p2 + p3)αZL
D−3
e2
∫dr
KD
2−1(p1r)KD
2−1(p2r)K∆−D
2+1(p3r)
Z(D2− 1)2Z(∆− D
2)r∆−1
+∆(D −∆− 2)
2
KD2−1(p1r)KD
2−1(p2r)K∆−D
2(p3r)
Z(D2− 1)2Z(∆− D
2)r∆−1
, (3.26)
and therefore holography gives (for transverse momenta):
〈Jx(p1)Jx(p2)O(p3)〉0 =αZL
D−3
e2Z(D2− 1)2Z(∆− D
2)
I(D2, D
2− 1, D
2− 1,∆− D
2+ 1)
+∆(D −∆− 2)
2I(D2− 1, D
2− 1, D
2− 1,∆− D
2
)δD(p1 + p2 + p3). (3.27)
– 18 –
Comparing with (2.30), we find complete agreement in the functional form, and fix the
coefficient
AJJO =αZL
D−3
e2Z(D2− 1)2Z(∆− D
2). (3.28)
3.1.4 ATTO
The computation of ATTO proceeds using similar tricks to the computation of AJJO.
However, due to the fact that the relevant correction to the bulk action is much higher
derivative, this computation is much more technically challenging. We leave the details
of this computation to Appendices B and C, and here only quote the main result:
ATTO = −LD−1
2κ2× 16αY
Z(D2
)2Z(∆− D2
). (3.29)
4 Holography: High Frequency Asymptotics
In this section, we reproduce (1.3) directly from holography. The derivation presented
in Section 2 does not rely on any “matrix largeN” limit, so it is natural to question what
we can learn from the holographic derivation. The key advantage of our holographic
formulation is that we will be able to derive the full asymptotic behavior of (1.3) non-
perturbatively in h. Even if the true ground state of the theory at finite h and T is
far from a CFT, the existence of a UV CFT is sufficient to impose (1.3). Holography
geometrizes this intuition in a very natural way – we will see that (1.3) is universally
independent of the low-energy details of the theory in these holographic models.
4.1 Scalar Two-Point Functions
As in Section 2, we begin by studying 〈O0(Ω)O0(−Ω)〉. To do so, we must perturb the
field ψ by a source at frequency Ω and compute the response, using (3.5) and (3.6).
We will begin by assuming that ψ = 0 identically on the background geometry. In this
case, ψ is the only field which will be perturbed in linear response, as there is no linear
in ψ term in (3.1). We will relax this assumption at the end of the derivation. We also
assume that ∆−D/2 is not an integer: this assumption is relaxed in Appendix A. For
now, we must solve the differential equation
∇a∇aψ = W (Φ)ψ, (4.1)
which leads to
1√g∂r (√ggrr∂rψ)− Ω2gttψ =
(∆0(∆0 −D)
L2+ αWL
D−52 Φ(r) · · ·
)ψ (4.2)
– 19 –
In general, the solution of this equation for all Ω requires knowledge of the full bulk
geometry.
However, if we restrict ourselves to studying asymptotic behavior at large Ω, a
great simplification occurs. To see this, it is helpful to rescale the radial coordinate to
R ≡ Ωr. (4.3)
We claim that the function ψ(R) is essentially non-vanishing when R ∼ 1. To see why,
note that (4.2) becomes
Ω2
√g∂R(√g grr ∂Rψ)− Ω2gttψ =
(∆0(∆0 −D)
L2+ αWL
D−52 Φ · · ·
)ψ. (4.4)
If this equation is only non-trivial when R ∼ 1, then perturbation theory about Ω =∞should be well-defined. Making use of the expansion for Φ (3.4), the right hand side of
(4.4) becomes(∆0(∆0 −D)
L2+αWL2
(h
ΩD−∆RD−∆ +
〈O〉(2∆−D)Ω∆
R∆
)+ · · ·
)ψ (4.5)
and so as Ω → ∞, we may neglect the scalar corrections to the equations of motion.
The Ω → ∞ limit is given by the r → 0 limit (keeping R fixed), and so we may
approximate the metric at leading order by (3.3). The left hand side of (4.4) is hence
Ω2
(R
ΩL
)D+1
∂R
((R
ΩL
)1−D
∂Rψ
)− Ω2
(R
ΩL
)2
ψ =R2∂2
Rψ + (1−D)R∂Rψ −R2ψ
L2
(4.6)
Corrections due to the deviation of the IR geometry from pure AdS (both from finite T
and finite h) are subleading for the same reason that W (Φ) corrections are subleading.
Indeed, comparing (4.5) and (4.6), we see that perturbation theory about Ω = ∞ is
well-controlled. Changing variables to
χ(R) ≡ R∆0−Dψ(R), (4.7)
and expanding χ as a perturbative expansion in small αW
χ = χ(0) + αWχ(1) + · · · . (4.8)
equation (4.4) reduces to
RD+2−∆0
[∂2Rχ
(0) +D + 1− 2∆0
R∂Rχ
(0) − χ(0)
]= 0. (4.9)
– 20 –
The regular solution to equation, conveniently normalized so that χ(0) = 1, is
χ(0)(R) =RbKb(R)
Z(b), (4.10)
where
b ≡ ∆0 −D
2. (4.11)
We begin by assuming that b > 0, which means that the O(R0) term in χ describes
the source, and the O(R2b) term in χ describes the response. We will discuss the case
of b < 0 in Appendix D.
At large but finite Ω, we must correct (4.9). The dominant contribution comes
from the scalar field corrections in (4.5), and so (4.9) becomes
∂2Rχ
(1) +1− 2b
R∂Rχ
(1) − χ(1) ≈ αWR2
(h
ΩD−∆RD−∆ +
〈O〉(2∆−D)Ω∆
R∆
)χ(0). (4.12)
Since Ω is large, the right hand side is perturbatively small and we may correct the
leading order solution (4.10) perturbatively. Keeping the boundary conditions χ(0) = 1
and χ(∞) = 0 fixed, the perturbative correction to (4.10) is unique, and it is straight-
forward to extract 〈O0O0〉. The perturbative computation of the finite Ω corrections
to this correlator proceed in a few simple mathematical steps.
First, we construct the Green’s function G(R;R0) for this differential equation:
∂2RG(R;R0) +
1− 2b
R∂RG(R;R0)−G(R;R0) = −δ(R−R0). (4.13)
Employing Bessel function identities including (3.24) and [32]
Ib(x)Kb−1(x) + Kb(x)Ib−1(x) =1
x, (4.14)
and our boundary conditions that G(0;R0) = G(∞;R0) = 0, we find:
G(R;R0) =
R0Kb(R0)Ib(R)(R/R0)b R < R0
R0Ib(R0)Kb(R)(R/R0)b R > R0. (4.15)
Using this Green’s function, we can readily construct the solution to the differential
equation
∂2Rχ+
1− 2b
R∂Rχ− χ = − c
Z(b)Ra+bKb(R), (4.16)
which is:
χ =c
Z(b)
∞∫0
dR0 G(R;R0)Ra+b0 Kb(R0). (4.17)
– 21 –
Keeping in mind our holographic application, we need only extract the O(R∆0) con-
tribution to ψ, or equivalently the O(R2b) contribution to χ. We now assume that
a+ 2 > 2b for simplicity – the opposite case is studied in Appendix D. In this case, the
leading order term as R→ 0 of the perturbation to χ is O(R2b):
χ(1)(R→ 0) ≈ c
Z(b)RbIb(R)
∞∫0
dR0 R1+aKb(R)2 =
c
Z(b)RbIb(R)Ψ(a+ 2; b). (4.18)
We now employ (4.18), using
(a, c) = (D −∆− 2,−αWh) or
(∆− 2,− αW 〈O〉
2∆−D
), (4.19)
to write down the leading order corrections to (4.12):
χ(R) = 1+R2b
[ CO0O0
2∆0 −D− αWh
22b−1Γ(b)Γ(1 + b)ΩD−∆Ψ(D −∆; b)
− αW 〈O〉22b−1Γ(b)Γ(1 + b)(2∆−D)Ω∆
Ψ(∆; b)
]+ · · · (4.20)
where CO0O0 is the holographic normalization of the operator O0, defined analogously
to COO in (3.11). Hence, our explicit non-perturbative calculation gives
〈O0(−Ω)O0(Ω)〉 = Ω2∆0−D
[CO0O0 −
αWΨ(D −∆; ∆0 − D2
)
22∆0−D−2Γ(∆0 − D2
)2
h
ΩD−∆
− αWΨ(∆; ∆0 − D2
)
22∆0−D−2Γ(∆0 − D2
)2(2∆−D)
〈O〉Ω∆
]+ · · · . (4.21)
It is simple to compare to the predictions of conformal perturbation theory. Combining
(2.14), (2.17), (3.11) and (3.16) and simplifying, we indeed obtain (4.21).
4.1.1 Subleading Orders in the Expansion
In the above derivation, we assumed that the geometry was AdS. In fact, the metric is
not quite AdS in the presence of a background ψ field: away from r = 0, there will be
deformations of the form
ds2 =L2
r2
(dr2 + dt2e + dx2
)+ O
(r2∆−2, r2(D−∆)−2, rD−2
). (4.22)
Because Einstein’s equations are sourced by gabψ2 and (∂ψ)2, the leading order terms
in the asymptotic expansion of (3.4) imply the first subleading corrections in (4.22).
– 22 –
These subleading corrections are small as r → 0 (or for R ∼ 1 as Ω→∞), and so they
will lead to perturbative corrections to 〈O0O0〉 of the form
〈O0(−Ω)O0(Ω)〉 = Ω2∆0−D[CO0O0 + · · ·+ #
h2
Ω2(D−∆)+ #〈O〉2Ω2∆
+ #h〈O〉ΩD
+ · · ·].
(4.23)
The last most term in (4.23), which leads to Ω−D corrections to the correlation function,
can likely be interpreted as corrections to 〈Tµν〉. Schematically, one finds [8]
〈O0(−Ω)O0(Ω)〉 = Ω2∆0−D[CO0O0 + · · ·+ CµνO0O0T
〈Tµν〉ΩD
+ · · ·]. (4.24)
The logic for such terms follows analogously to the appearance of the B-term in (1.3)
– the stress tensor will appear in the O0O0 OPE, and so terms proportional to pres-
sure and energy density will appear in the asymptotic expansion of generic correlation
functions. For instance, this was noted for the conductivity [8, 13] and shear viscosity
[11–13, 17].
There are other types of corrections that can arise, which are rather straightforward.
In all cases, the asymptotic expansion of the correlation function in powers of 1/Ω
is cleanly organized by the behavior of the bulk fields near the AdS boundary. The
perturbative derivation that we described earlier straightforwardly accounts for all such
corrections.
4.2 Conductivity
Next, we compute the asymptotics of σ(Ω) at large Ω. The structure of the computation
is very similar to the computation of the asymptotics of 〈O0(Ω)O0(−Ω)〉. We compute
within linear response the fluctuations of the gauge field Ax and employ the asymptotic
expansion (3.5) to compute 〈Jx〉/ΩA0x. Since our background is uncharged, if we turn
on a source for Ax, no other bulk fields will become excited (this is a consequence of
the fact that Ax is a spin 1 mode under the spatial SO(D−1) isotropy). The equations
of motion then reduce to
0 =1√g∂a (√gZ(Φ)F ax) =
1√g∂r (√gZ(Φ)grrgxx∂rAx)− Ω2Z(Φ)gttgxxAx. (4.25)
Upon the variable change (4.3), and keeping only finite Ω corrections from the scalar
field Φ (for the same reasons as we mentioned in the previous section), we find that the
leading order corrections to (4.25) at large but finite Ω can be found by perturbatively
solving
∂2RAx +
3−DR
Ax − Ax ≈ −∂Ra(R) ∂RAx (4.26)
– 23 –
where
a(R) = αZ
[h
ΩD−∆RD−∆ +
〈O〉(2∆−D)Ω∆
R∆
]. (4.27)
We see that this differential equation is extremely similar to the one studied in the
previous subsection, upon replacing χ with Ax and setting (for the remainder of this
section)
b =D
2− 1. (4.28)
The zeroth-order solution at Ω = ∞ is given by (4.10), and the Green’s function for
the left hand side of (4.26) is given by (4.15). The only difference is that the source,
which we must integrate over to recover the subleading behavior of Ax, is a bit more
complicated. Again, we begin by assuming that
a(R) = cRa. (4.29)
The coefficient is chosen to simplify the equations in the remainder of the paragraph.
The boundary conditions on Ax are Ax(0) = 1, Ax(∞) = 0. At first order in c, we find
A(1)x =
c
Z(b)
∞∫0
dR0 G(R;R0)[R2b−1
0 ∂R0
(R1+a−2b
0 ∂R0Kb(R0))−Ra
0Kb(R0)]. (4.30)
Now, using
∂2xKb(x) =
2b− 1
x∂xKb(x) + Kb(x), (4.31)
we may simplify the integral in square brackets in (4.30) to
A(1)x =
c
Z(b)
∞∫0
dR0 G(R;R0)[aRa−1
0 ∂R0Kb(R0)], (4.32)
As in the subsection above after (4.18), let us assume that a is large enough that the
following integration by parts manipulations are acceptable:
∞∫0
dR0 KbRa−2b0 ∂R0Kb(R0) =
∞∫0
dR0 Ra−2b0
1
2∂R0
(Kb(R0)
)2
≈(b− a
2
) ∞∫0
dR0 Ra−1−2b0 Kb(R0)2
=(b− a
2
)Ψ(a; b). (4.33)
– 24 –
Hence, we find (for a single power a)
A(1)x (R→ 0) ≈ c
Z(b)Ib(R)Rb × a
(b− a
2
)Ψ(a; b). (4.34)
Keeping in mind that the true source is (4.27), we conclude that
Ax(R→ 0) ≈ Kb(R)
Z(b)− R2b
2b
αZh
ΩD−∆
D −∆
Z(b)2
(1− ∆
2
)Ψ(D −∆; b)
− R2b
2b
αZ〈O〉(2∆−D)Ω∆
∆
Z(b)2
(1− D −∆
2
)Ψ(∆; b). (4.35)
Hence,
σ(iΩ) =LD−3
e2ΩD−3
[σ∞ −
αZh
ΩD−∆
D −∆
Z(D2− 1)2
(1− ∆
2
)Ψ(D −∆; D
2− 1)
− αZ〈O〉(2∆−D)Ω∆
∆
Z(D2− 1)2
(1− D −∆
2
)Ψ(∆; D
2− 1)]. (4.36)
It is simple to compare to the predictions of conformal perturbation theory. Combining
(2.32), (3.11) and (3.28) and simplifying, we indeed obtain (4.36).
4.3 Viscosity
The computation of the viscosity proceeds similarly to the previous two subsections.
The SO(D−1) isotropy of the background implies that gxy will not couple to any other
modes. We hence write the perturbation to the metric as
δ(ds2) = 2L2
r2hxydxdy, (4.37)
and find the differential equation governing hxy. We do this computation in Appendix
B, and here quote the leading order answer (at finite Ω):(L
r
)D−1 [∂2rhxy −
D − 1
r∂rhxy − r2Ω2hxy
]=
8
D − 1∂r
[(L
r
)D−3
Y (Φ)2(D − 1)Ω2∂rhxy
]
− 8
D − 1
(L
r
)D−3
Y (Φ)((D − 2)Ω4hxy + Ω2∂2
rhxy)
+8
D − 1∂2r
[(L
r
)D−3
Y (Φ)(Ω2hxy + (D − 2)∂2
rhxy)]. (4.38)
– 25 –
After the variable change (4.3), we obtain
∂2Rhxy −
D − 1
R∂Rhxy −R2hxy =
8
D − 1
(L
R
)−2
Y (Φ)((D − 2)hxy + ∂2
Rhxy)
− 8
D − 1
(R
L
)D−1
∂R
[(L
R
)D−3
Y (Φ)2(D − 1)∂Rhxy
]
+8
D − 1
(R
L
)D−1
∂2R
[(L
R
)D−3
Y (Φ)(hxy + (D − 2)∂2
Rhxy)]. (4.39)
We now proceed similarly to the previous subsection. When Ω → ∞, Y vanishes and
(4.39) is solved by (4.10) with
b =D
2. (4.40)
Since Y (Φ) will be a sum of two power laws in R, to first order in perturbation theory
we can determine hxy by solving (4.39) after replacing Y (Φ) with L2a(R), with a(R)
as defined in (4.29). At first order in c, assuming convergence of integrals, we find
hxy(R→ 0) ≈ −8c
D − 1Ib(R)Rb
∫dR0Kb(R0)
[Ra+3−2b
0
((D − 2)Kb(R0) + ∂2
R0Kb(R)
)− 2(D − 1)∂R0
(Ra+3−2b
0 ∂R0Kb(R))
+ ∂2R0
+ ∂2R0
(Ra+3−2b
0
(Kb(R0) + (D − 2)∂2
R0Kb(R)
))]. (4.41)
Repeated application of the identities (2.13), (4.18), (4.31) and (4.33) leads to
hxy(R→ 0) = −8cIb(R)Rb[a(a+ 1)Ψ(a+ 2; b) +
a
2(2b− a)(2b− 2)(a+ 1)Ψ(a; b)
]= −8cIb(R)Rb × a
4(2b− a)(a+ 2)(2b− a− 2)Ψ(a; b) (4.42)
Now, we have
(a, c) = (D −∆, αY h) or
(∆,
αY 〈O〉2∆−D
), (4.43)
so we conclude that
η =LD−1
2κ2ΩD−1
[CTT −
2αY h
Z(D2
)2∆(D −∆)(D −∆ + 2)(∆− 2)Ψ
(D −∆;
D
2
)1
ΩD−∆
− 2αY 〈O〉(2∆−D)Z(D
2)2
∆(D −∆)(∆ + 2)(D −∆− 2)Ψ
(∆;
D
2
)1
Ω∆
](4.44)
Comparing this equation to (2.40), (3.11) and (3.29), we again find that our holographic
answer agrees with conformal perturbation theory.
– 26 –
5 Holography: Full Frequency Response
A natural advantage that holography provides to quantum field theories is the capa-
bility of directly exploring the real time response functions at all frequencies. In this
section, we investigate the holographic linear response to scalar deformations of the
CFT. Following [19], we account for the thermal expectation values 〈O〉T ∼ T∆ that
generally arise in CFTs. Using this model, we can solve the equations of motion for
the dynamical fields at all frequencies and calculate the response functions studied in
the previous sections.
5.1 A Minimal Model
We will study holographic models of the form (3.1), for simple choices of W , V , Z
and Y . Essentially, we will truncate the asymptotic expansions of these Φ-dependent
functions at lowest non-trivial order:
V (Φ) =− D(D − 1)
2L2κ2+
∆(D −∆)
2L2Φ2 (5.1a)
Z(Φ) =1 + αZLD−1
2 Φ (5.1b)
Y (Φ) =αYLD+3
2 Φ (5.1c)
W (Φ) =∆0(D −∆0)
L2+ αWL
D−52 Φ (5.1d)
Using these simple couplings, we will study the finite temperature response both for
ω T and ω . T .
Assuming that κ → 0, we may solve for the background geometry without con-
sidering fluctuations of the matter content: Aµ, Φ or ψ. We will focus on the planar
AdS-Schwarzchild black hole solution1 to the resulting equations of motion, whose real-
time metric reads [10]:
ds2 =
(4πT
D
)2L2
u2(−f(u)dt2 + ηijdx
idxj) +L2du2
u2f(u), (5.2)
where f(u) = 1 − uD and T is the Hawking temperature of the black hole, as well as
the temperature of the dual field theory. The coordinate u is dimensionless: u = 0
corresponds to the AdS boundary, while u = 1 corresponds to the black hole horizon.
In the asymptotically AdS regime (u → 0), the coordinate u is a simple rescaling of
the coordinate r in (3.3): u = 4πTr/D.
1We consider other background metrics in Section 5.3.1.
– 27 –
We now study the perturbative fluctuations of the matter fields Φ, Aµ and ψ about
this background solution. Varying (3.1) with respect to Φ, we obtain:
0 = (∇2 −m2)Φ +αYL
D+32
κ2CabcdC
abcd. (5.3)
Because the black hole background (5.2) is translation invariant in the boundary direc-
tions, CabcdCabcd depends only on u, and thus we may look for a solution of the form
Φ(u). The resulting ordinary differential equation is
0 = Φ′′(u)+
(f ′(u)
f(u)−D − 1
u
)Φ′(u)+
∆(D −∆)
u2f(u)Φ(u)+
D(D − 1)2(D − 2)αY LD−1
2 u2D−2
κ2f(u)(5.4)
This equation can be solved using standard Green’s function techniques analogous to
the previous section. We find
Φ(u) = 2F1
(∆D, ∆D
; 2∆D
;uD) (
Φ1 −D(D − 1)2(D − 2)αYL
D−12
(2∆−D)κ2g∆(u)
)u∆
+ 2F1
(D−∆D, D−∆
D; 2D−∆
D;uD
) (Φ0 +
D(D − 1)2(D − 2)αYLD−1
2
(2∆−D)κ2h∆(u)
)uD−∆ ,
(5.5)
where Φ0 and Φ1 are integration constants, 2F1(z1, z2 ; z3 ; z4) denotes the standard
hypergeometric function, and g∆(u) and h∆(u) are dimensionless functions given by2
g∆(u) =
u∫0
dy y2D−1−∆2F1
(D−∆D, D−∆
D; 2D−∆
D; yD
), (5.6a)
h∆(u) =
u∫0
dy yD−1+∆2F1
(∆D, ∆D
; 2∆D
; yD). (5.6b)
Note that g∆(0) = h∆(0) = 0. Hence, at the AdS boundary, the asymptotic behavior
of Φ(u) is
φ(u) = Φ0uD−∆
(1 + O(uD)
)+ Φ1u
∆(
1 + O(uD)). (5.7)
2This representation of the solution is only valid for ∆ < 2D. In particular, the integral defining
g∆(u) in (5.6) diverges for ∆ ≥ 2D. Further, the two independent solutions presented in (5.5) are
actually identical for ∆ = D/2. Of course, the coefficients of g∆(u) and h∆(u) also diverge for this
particular value of ∆. However, we note that the conductivity is still a smooth function of ∆ at this
special value and so where results are presented for ∆ = D/2, we have added a small positive number
to the scaling dimension: ∆ = D/2 + ε where ε ∼ 10−7.
– 28 –
Figure 2: The scalar field with D = 4 as a function of u for various scaling dimensions
∆ with Φ1 fixed to 1.
Φ0 and Φ1 encode the source h and response 〈O〉 respectively, as we discussed previously
in (3.5). However, since we are using the dimensionless radial coordinate u here, it is
important to note that there will be additional powers of T relating the physical source
and response with the coefficients Φ0,1:
Φ1 =〈O〉
(2∆−D)LD−1
2
(D
4πT
)∆
; Φ0 =h
LD−1
2
(D
4πT
)D−∆
. (5.8)
The integration constant Φ1 is fixed by demanding regularity at the black hole horizon:
Φ1 =− Φ0
Γ(2− 2∆
D
)Γ(
∆D
)2
Γ(1− ∆
D
)2Γ(
2∆D
)+αYL
D−12 D(D − 1)2(D − 2)
κ2(2∆−D)
(g∆(1)− Γ
(2− 2∆
D
)Γ(
∆D
)2
Γ(1− ∆
D
)2Γ(
2∆D
)h∆(1)
).
(5.9)
Note that g∆(1) and h∆(1) are finite and can be determined numerically. Some sample
plots of Φ(u), upon setting h = 0, are given in Figure 3.
Due to the C2 coupling in this holographic model, the scalar field Φ spontaneously
acquires an expectation value upon subjecting the CFT to a finite temperature T :
〈O〉T ∼ Φ1T∆. At finite detuning h, 〈O〉 picks up a simple linear correction in h, as
can be seen from (5.9). This is an artifact of the ‘probe’ limit where we have neglected
the backreaction of Φ on the metric. Despite this limitation, this holographic system is
a useful way of modeling the finite temperature response of a CFT at all frequencies.
– 29 –
5.2 Scalar Two-Point Functions
In this section we will construct the full frequency response for the minimal model (3.1)
and show that the analysis done in Section 4.1 accurately predicts the high-frequency
behavior of the scalar two point function 〈O0O0〉. Given the explicit model introduced
in Section 5.1, the equation of motion for ψ is
0 =
(∇2 +
∆0(D −∆0)
L2+ αWL
D−52 Φ
)ψ
=1√−g∂u(
√−gguu∂uψ)− ω2gttψ +∆0(D −∆0)
L2ψ + αWL
D−52 Φψ .
(5.10)
As before, ψ is only dependent on u; the resulting ordinary differential equation reads
0 = ψ′′ +
(f ′
f− D − 1
u
)ψ′ +
ω2D2ψ
(4πT )2f 2+
∆0(D −∆0)
u2fψ +
αWLD−1
2 Φψ
u2f. (5.11)
Because this equation is homogeneous, one of the two boundary conditions we impose is
‘arbitrary’: the observable quantity we are extracting is the ratio of 〈O0〉/h0, as in (3.6),
and this is unaffected under rescaling: ψ → λψ. The boundary conditions to (5.11)
require some care [10]. Near the horizon, ψ(u) acquires a log-oscillatory divergence:
this is a consequence of the fact that we require that matter must be ‘infalling’ into
the black hole. In order to regulate the divergence, we will introduce another field
ψ(u) = f(u)bΨ(u) with b = −iω/4πT , and enforce regularity on Ψ(u) via the following
mixed boundary condition:
Ψ′(1) =Ψ(1)
D(2b+ 1)
(∆0(D −∆0) + αWL
D−12 Φ(1)
). (5.12)
Numerically, we solve this equation of motion by ‘shooting’ any solution obeying (5.12)
from the black hole horizon to the asymptotic boundary, and obtain 〈O0O0〉 from the
resulting asymptotic behavior. We know from (3.5) that ψ(u) = ψ0uD−∆ +ψ1u
∆ + · · ·as u → 0. Hence, we use regression on points sampled near the boundary to compute
ψ0 and ψ1. Using (3.4) and (3.6), which are valid for arbitrary asymptotically AdS
geometries, we numerically obtain the scalar two point function, which is plotted in
Figure 3.
5.3 Conductivity
In this section we calculate the full frequency-dependent conductivity for the model of
Section 5.1, and confirm the high-frequency analysis done in Section 4. The computa-
tion proceeds similarly as in the previous section. In the u coordinate, the conductivity
– 30 –
Figure 3: Scalar two point function 〈O0(−Ω)O0(Ω)〉 in D = 3 at finite temperature,
with scaling dimensions ∆0 = 2 and ∆ = 2. We have set αY αW = 1. Left: full
frequency response for the scalar two point function. Right: comparison between the
analytic prediction for the leading correction to the linear response plotted against the
numerically calculated response with the leading term removed (plotted in a log-log
plot).
is given by solving the equation of motion for Ax, assuming implicit frequency depen-
dence of the form e−iωt:
A′′x(u) +
(Z ′(u)
Z(u)+f ′(u)
f(u)− D − 3
u
)A′x(u) +
(Dω
4πT
)2Ax(u)
f(u)2= 0 , (5.13)
where Z(u) ≡ Z(Φ(u)). Just as in the previous subsection, we must solve (5.13) with
an appropriate infalling boundary condition. As before, we write Ax(u) = f(u)bF (u);
the regularity condition for F (u) at the horizon is
F ′(1) = − bF (1)
1 + 2b
(2 +
Z ′(1)
Z(1)
). (5.14)
We employ the same shooting method in order to construct our solution, and hence
extract σ(ω).
We extract σ(ω) through the asymptotic behavior via (3.5). Assuming Ax ∼ A0 +
· · ·+ A1uD−2, and D odd, the conductivity is given by
σ(ω) = −LD−3
e2
(D − 2)A1
iωA0
(4πT
D
)D−2
. (5.15)
When D is even, the presence of logarithms in the asymptotic expansion of Ax(u)
complicates the story. Let us simply note the proper prescription for this case. Given
the asymptotic expansion [33]
Ax(u) = A0
[1 + 8b2u2 log(Λu)
]+ A1u
2 + · · · (5.16)
– 31 –
Figure 4: The Euclidean (left) and real (right) AC conductivity for D = 3 and ∆ = 2
in the critical theory (h = 0) for various choices of interaction strength αWαZ .
one finds
σ(ω) = −π2T 2L
iωe2
(2A1
A0
− ω2
2
). (5.17)
Note that the logarithm in (5.16) is analogous to the logarithm that arose in (2.26),
within conformal field theory. It will not affect the real part of σ(ω).
Let us note in passing that we may compute the DC conductivity σ(ω = 0) ana-
lytically via the ‘membrane paradigm’ [34, 35]:
σ(0) =
(4πLT
D
)D−3Z(u = 1)
e2. (5.18)
As expected, the full numerical solution reproduces this result.
Figures 4 and 5 show σ(ω) in D = 3 and D = 4, respectively, for varying coupling
constants αWαZ . This is analogous to changing the CFT data CJJO. Not surprisingly,
we find that the non-trivial structure in the conductivity becomes enhanced as this
coupling strength increases.
As we noted after (1.4), there exist sum rules which tightly constrain the real part
of the conductivity whenever h = 0, so long as ∆ > D − 2. In particular, when such a
sum rule is satisfied, it implies that if the conductivity is enhanced at high ω, it must
be suppressed at low ω, or vice versa. Figure 6 shows the dependence of the real part of
σ(ω) on the dimension of the operator O in D = 3 and D = 4. We clearly observe that
the conductivity is more affected by relevant operators of smaller dimension. When
the sum rules are violated, we observe dramatic enhancement of the conductivity at
all frequencies. In D = 4, we note that an operator of dimension ∆ = 1, leads to a
constant shift in the real part of the conductivity at high frequency, which is cleanly
observed in the figures 5 and 6.
– 32 –
Figure 5: The real part of the AC conductivity for D = 4 and ∆ = 1 (left) and ∆ = 3
(right), in the critical theory (h = 0), for various choices of coupling constants αWαZ .
At large ω, Re(σ(ω)) ≈ σ∞ω, with σ∞ = π2CJJ given in (2.28).
Figure 6: The real frequency AC conductivities for D = 3 (left) and D = 4 (right)
for various choices of scaling dimension ∆, with fixed αWαZ = 0.1. For D = 3 the sum
rules are observed for ∆ > 1.0 and for D = 4 the sum rules are observed for ∆ > 2.0.
Significant enhancement of Re(σ) is observed whenever the sum rules are violated.
Figure 7 compares the analytic predictions for the asymptotic approach of σ(ω) to
its CFT value from Section 4 to our numerical calculation. We find excellent agreement.
We observe that the figure numerically confirms that when the scalar deformation is
critical (Φ0 = 0) the high-frequency conductivity is proportional to ΩD−3−∆, while
when the scalar is tuned away from the critical point (Φ0 6= 0) the high-frequency
conductivity is proportional to Ω∆−3.
5.3.1 Reissner-Nordstrom Model
So far, the background geometry has been completely thermal. As a consequence, we
have not checked the validity of our asymptotic results when 〈O〉 depends nonlinearly
– 33 –
Figure 7: A comparison of our numerical calculation of σ(iΩ) (shown in dots) to the
first subleading contribution in the asymptotic expansion (1.3) (shown as thin solid
line), when Ω T . We take ∆ = 2 and D = 3. As predicted in (1.3), when h = 0
the first subleading power Ω−∆ is smaller than when h 6= 0. For h 6= 0, the asymptotic
corrections are proportional to h and governed by the same power Ω∆−D, leading to a
slower decay.
on non-thermal detuning parameters.
To perform such a check, we now consider the real time action
S =
∫dD+1x
√−g(R
2κ2− Z(Φ)
F 2
4e2− 1
2(∂Φ)2 − V (Φ)
), (5.19)
with Z and V given in (5.1). Compared to (3.1), we have removed the ψ field as well
as the ΦC2 coupling. As before, let us imagine that the backreaction of Φ on the other
matter fields can be ignored, but let us now deform the CFT by a finite temperature T
and a finite charge density ρ. The dual geometry to such a theory is known analytically
for any density [10], and is called the AdS-Reissner-Nordstrom black hole:
ds2 =L2
r2+u
2
(−f(u)dt2 + δijdxidxj
)+L2du2
u2f(u)(5.20a)
A =
√D − 1
D − 2
e r0ξ
κL(1− uD−2)dt (5.20b)
where
f(u) = 1− (1 + ξ2)uD + ξ2u2(D−1) (5.21)
– 34 –
and r+ and ξ are related to the charge density ρ and temperature T of the black hole:
T =D − (D − 2)ξ2
4πr+
, (5.22a)
ξ2 =κ2e2ρ2r2D−2
+
(D − 1)(D − 2)L2D−4. (5.22b)
The dimensionless radial coordinate u is once again chosen in such a way that the
“outer” black hole horizon is located at u = 1 and the asymptotic AdS boundary is
located at u → 0. The black hole horizon is extremal when T = 0: this occurs at
precisely ξ2 = DD−2
, and so we will restrict our attention to ξ2 < DD−2
.
Around this background geometry, we can compute Φ(u), whose equation of motion
is given by
(∇−m2)Φ− αZLD−1
2
4e2FabF
ab = (∇−m2)Φ +αZL
D−12
2e2
u4A′2tr2
+
= 0. (5.23)
When ξ 6= 0, we will need to solve this equation numerically to determine Φ(u).
Due to the presence of the finite charge density, the equations of motion for Axbecome more complicated within linear response [10]. In particular, they couple to
fluctuations in the metric components gtx and grx. Using standard techniques, we are
able to reduce the equations of motion to a ‘massive’ equation of motion for Ax:
0 = A′′x +
(Z ′
Z+f ′
f− D − 3
u
)A′x +
ω2r2+
f 2Ax −
2(D − 1)(D − 2)u2ξ2
fAx. (5.24)
This equation is identical to that in [10], up to the dependence on Z(Φ), and it
can be evaluated numerically. Imposing infalling boundary conditions requires writ-
ing Ax(u) = f(u)bF (u) as before; the boundary condition at the black hole horizon
imposing regularity is
F ′(1)
F (1)=
1
(1 + 2b)f ′(1)
(2(D − 1)(D − 2)ξ2 − bf ′(1)
(Z ′(1)
Z(1)+f ′′(1)
f ′(1)− D − 3
u
)).
(5.25)
Figure 8 demonstrates agreement between our analytic prediction (1.3) for the high
frequency corrections to σ(ω), and a full numerical computation. It is important to
keep in mind that in this more generic geometry, the conductivity is not given by the
CFT result even when h = T = 0. Indeed, due to the presence of a finite charge density
ρ, and hence chemical potential µ, the conductivity takes the following form:
σ(ω) = σ∞ · (iω)D−3
[1 +A h
(iω)D−∆+ B 〈O〉
(iω)∆−A′µ
2
ω2+ · · ·
]. (5.26)
– 35 –
Figure 8: Comparison of Re[σ(ω) − σ∞] between our analytic prediction (1.3) for
the subleading correction to the conductivity, and the full dynamical conductivity at
αY αZ = ξ = 1 in D= 3. For simplicity we have set h = 0. As in Fig. 7, we observe
excellent agreement when ω T ; at low frequency, the structure of σ(ω) is highly
non-trivial in this background: see Figure 9.
The coefficients A and B are given in (2.32) and (4.36). The coefficient A′ is related to
the four-point correlation function 〈JJJJ〉 in the CFT, which is non-vanishing in our
holographic model due to graviton exchange in the bulk. The A′ term is in fact the
counterpart of the higher order correction (h/ωD−∆)2, where the role of O is played by
the charge density Jt, and the chemical potential µ plays the role of h. In (5.26), we
see that if h = 0, it is necessary to deform this field theory at finite charge density by
a relevant operator of dimension ∆ < 2 in order for the expansion (1.3) to hold.
In Figure 9 we plot Re(σ(ω)) as a function of αWαZ , corresponding to increasing
CJJO. Unlike before, in this field theory at finite charge density we observe non-trivial
structure even when αWαZ = 0, when the scalar operator O is decoupled. This is a
consequence of the fact that finite µ already corresponds to a non-trivial deformation
of the CFT. Upon setting αWαZ > 0, we observe additional structure emerge in Figure
9, associated with the coupling to the operator O.
In Figure 9, one may notice the presence of “missing” spectral weight. This spectral
weight has in fact shifted into a δ-function at ω = 0: ‘hydrodynamics’ constrains the
low frequency conductivity to be [10]
σ(ω → 0) =ρ2
ε+ P
(πδ(ω) +
i
ω
)+ · · · . (5.27)
Here ε is the energy density and P is the pressure. A uniform and static electric field can
impart momentum into a system at finite charge density at a fixed rate via the Lorentz
– 36 –
Figure 9: The real part of the conductivity for D = 3, ∆ = 2 and ‘charge parameter’
ξ = 1 for various values of αZαW . Coupling to the relevant operator O leads to
pronounced features in Re(σ) at intermediate frequencies, in addition to modifying the
high frequency asymptotics as in (5.26).
force: this is the origin of this δ function. This δ function must be taken into account
when evaluating sum rules, and upon doing so, the sum rules for the conductivity are
restored.
6 Lifshitz Holography
In this section, we will now extend our analysis to holographic quantum critical systems
with dynamic critical exponent z 6= 1. This means that there is an emergent scale
invariance of the low energy effective theory, but time and space scale separately. Under
a rescaling λ:
x→ λx, t→ λzt. (6.1)
Because time and space are no longer related through Lorentz transformations, in this
section we will not talk about the spacetime dimension D, but instead the spatial
dimension d = D − 1, as is more conventional in condensed matter.
As argued in [20], the high frequency behavior of correlation functions in such
Lifshitz theories will share many features with CFTs. For simplicity, we will only talk
about the conductivity in this section. In particular, the asymptotic expansion of the
conductivity will be modified from (1.3) to
σ(iΩ) = Ωd−2z
[CJJ +
AhΩ1+ d−∆
z
+B〈O〉Ω
∆z
+ · · ·]. (6.2)
– 37 –
We now present a holographic Lifshitz theory where this result can be shown explicitly,
and a relationship between A and B can be obtained.
6.1 Bulk Model
We first begin by outlining the holographic model which we study. The bulk action is
given by
S = Sbg[gµν ,Φ] +
∫dd+2x
√−gZ(Φ)
4e2FabF
ab, (6.3)
with Sbg the action for the metric and the other bulk fields which support the Lifshitz
geometry, and the remaining term describing a fluctuating gauge field. We take Z(Φ)
to be given by (3.2b), as before. Specific forms of Sbg can be found in [36], for example,
but they will not be relevant for our purpose. We will take our background to be
uncharged under Fab, the bulk field under which we will compute the conductivity.
The quadratic terms in Φ in Sbg are given by
Sbg =
∫dd+2x
√−g(
1
2(∂Φ)2 +
∆(∆− d− z)
L2Φ2
)+ · · · . (6.4)
Note that the mass of Φ2 is distinct from that in (3.2d), once z 6= 1.
The background metric of the Lifshitz theory is given by [36]
ds2 =L2
r2
[dr2 − dt2
r2(z−1)+ dx2
]. (6.5)
The isometries of this metric map on to the Lifshitz symmetries of the dual field the-
ory, and as a consequence the correlation functions of the dual theory are necessarily
Lifshitz invariant. As Lifshitz invariance is a much weaker requirement than conformal
invariance [37–39], it is not clear that holographic models can capture the dynamics of
generic Lifshitz theories. Nonetheless, we will be able to demonstrate a generalization
of (1.3) in at least one class of interacting Lifshitz theories, and although our specific
results for A and B are not generic, we expect that the qualitative features of this
model are more robust. One can also study more complicated bulk models than (6.3),
as the symmetries of Lifshitz theories are far less constraining [40].
In order to compute the conductivity, we must note changes to the holographic
dictionary in a Lifshitz background. Following the discussion of holographic renormal-
ization in [10], one finds:
Φ =h
Ld/2rd+z−∆ +
〈O〉(2∆− d− z)Ld/2
r∆ + · · · , (6.6a)
Ax = A0x +
e2
Ld−2
〈Jx〉d+ z − 2
rd+z−2 + · · · . (6.6b)
– 38 –
6.2 Asymptotics of the Conductivity
The equations of motion for the gauge field in the background (6.5) are
0 =1√g∂a (Z
√gF ax) = rd+z+1∂r
(Z(Φ(r))r1−d−z∂rAx
)− Ω2Z(Φ(r))r2zAx. (6.7)
Let us now define the variable
R ≡ Ω
zrz. (6.8)
By the same logic as before, the limit Ω→∞ becomes completely regular, with (6.7)
becoming
0 = Rd−2z ∂R
(Z
((zR
Ω
)1/z)R−
d−2z ∂RAx
)− Z
((zR
Ω
)1/z)Ax (6.9)
At Ω =∞, the solution to this equation with the correct boundary conditions is
Ax =K d+z−2
2z(R)
Z(d+z−2
2z
) . (6.10)
At finite Ω, we must solve
∂2RAx −
d− 2
zR∂RAx − Ax = a(R)Ax −R
d−2z ∂R
(a(R)R−
d−2z ∂RAx
)(6.11)
perturbatively in a(R). This is exactly analogous to the solution of (4.26). Following
(4.34), and using that
a(R) = αZh
(zR
Ω
) d+z−∆z
+αZ〈O〉
2∆− d− z
(zR
Ω
)∆z
, (6.12)
we obtain (analogously to (4.35)):
Ax(R→ 0) ≈K d+z−2
2z(R)
Z(d+z−2
2z
) +R
d+z−2z
d+z−2z
×αZh
( zΩ
) d+z−∆z (d+ z −∆)(∆− 2)
2z2Z(d+z−22
)2Ψ
(d+ z −∆
z,d+ z − 2
2z
)
+αZ〈O〉
2∆− d− z( z
Ω
)∆z ∆(d+ z −∆− 2)
2z2Z(d+z−22z
)2Ψ
(∆
z,d+ z − 2
2z
). (6.13)
Hence, the conductivity is
σ(iΩ) =Ld−2
e2Ω
d−2z
[CJJ +
αZz2−z−∆
z (d+ z −∆)(∆− 2)
2Z(d+z−22z
)2Ψ
(d+ z −∆
z,d+ z − 2
2z
)h
Ωd+z−∆
z
+αZz
2−d−2z+∆z ∆(d+ z −∆− 2)
2(2∆− d− z)Z(d+z−22z
)2Ψ
(∆
z,d+ z − 2
2z
) 〈O〉Ω
∆z
]. (6.14)
– 39 –
6.3 Three-Point Function and (Holographic) Lifshitz Perturbation Theory
Now, we follow the construction of Section 3.1.3 and compare our holographic result
to “Lifshitz perturbation theory”.
The properly normalized solution to the scalar equation of motion in a Lifshitz
background that follows from (6.4),
∆(∆− d− z)
L2Φ = ∇a∇aΦ, (6.15)
is
Φ(p3, r) = eip3·xrd+z−∆K 2∆−d−z
2z(ωzrz)
Z(2∆−d−z2z
)Ld/2jO(p3). (6.16)
The cubic contribution to the bulk action becomes
Sbulk =
∫dd+2x
√gαZL
d/2Φ(p3)
e2gabgxx∂aAx(p1)∂bAx(p2)
= −∫
dd+2xAx(p1)∂a
[√gαZL
d/2Φ(p3)
e2gabgxx∂bAx(p2)
]= −
∫dd+2xAx(p1)
√gαZL
d/2
e2gabgxx∂bAx(p2)∂aΦ(p3)
=αZL
d/2
2e2
∫dd+2x
√gAx(p1)Ax(p2)
[gxx
∆(∆− d− z)
L2Φ(p3) +
2
rgxxgrr∂rΦ(p3)
].
(6.17)
Using (6.10) and (6.16), along with (3.24) we obtain (suppressing the δ-function for
momenta, and the integrals in the boundary directions)
Sbulk =αZL
d−2
2e2
∫dr
rd+z−1
K d+z−22z
(p1
zrz)
Z(d+z−2
2z
) K d+z−22z
(p2
zrz)
Z(d+z−2
2z
) [∆(∆− d− z)
K 2∆−d−z2z
(p3
zrz)
Z(2∆−d−z2z
)
+2∆K 2∆−d−z
2z(p3
zrz)
Z(2∆−d−z2z
)−
2zK 2∆−d+z2z
(p3
zrz)
Z(2∆−d−z2z
)
]rd+z−∆ (6.18)
Changing variables to ρ = rz/z, we obtain
Sbulk =αZL
d−2
2e2
∫dρ(zρ)
2−z−∆z
Z(d+z−22z
)2Z(2∆−d−z2z
)
[∆(∆ + 2− d− z)K 2∆−d−z
2z(p3ρ)
−2zK 2∆−d+z2z
(p3ρ)]
K d+z−22z
(p1ρ)K d+z−22z
(p2ρ). (6.19)
– 40 –
Analogously to (3.27), we obtain
〈Jx(Ω1)Jx(Ω2)O(Ω3)〉0
=αZL
d−2z2−z−∆
z
e2Z(d+z−22z
)2Z(2∆−d−z2z
)
zI
(d+ z
2z,d+ z − 2
2z,d+ z − 2
2z,2∆− d+ z
2z
)−∆
2(∆ + 2− d− z)I
(d− z
2z,d+ z − 2
2z,d+ z − 2
2z,2∆− d− z
2z
). (6.20)
We now attempt to follow the arguments of conformal perturbation theory. We
expect to find that the conductivity takes an analogous form to (1.3):
σ(iΩ) = Ωd−2z
[σ∞ +
AhΩ
d+z−∆z
+B〈O〉
Ωd+z−∆
z
], (6.21)
and now use the form of the three-point correlator (6.20) to fix A and B. We take p1,2,3
to be given by (2.7), and Taylor expand in p/Ω. We find a regular term as p→ 0, given
by
〈Jx(Ω)Jx(−Ω)O(0)〉0 =αZL
d−2z2−z−∆
z Ω∆−z−2
z
e2Z(d+z−22z
)2Z(2∆−d−z2z
)
zZ(
2∆− d+ z
2z
)−∆
2(∆ + 2− d− z)Z
(2∆− d− z
2z
)Ψ
(d+ z −∆
z,d+ z − 2
2z
)= −αZL
d−2z2−z−∆
z Ω∆−z−2
z
e2Z(d+z−22z
)2
(1− ∆
2
)(d+ z −∆)Ψ
(d+ z −∆
z,d+ z − 2
2z
).
(6.22)
Similarly, we find a non-analytic contribution in p:
〈Jx(Ω)Jx(−Ω)O(p)〉0 = · · · − αZLd−2z
2−z−∆z Ω
∆−z−2z
e2Z(d+z−22z
)2Z(2∆−d−z2z
)×
∆(∆ + 2− d− z)
2Z(d+ z − 2∆
2z
)Ψ
(∆
z,d+ z − 2
2z
)( pΩ
) 2∆−d−zz
. (6.23)
Following the logic of conformal perturbation theory, we fix
A = −αZLd−2z
2−z−∆z
e2Z(d+z−22z
)2
(1− ∆
2
)(d+ z −∆)Ψ
(d+ z −∆
z,d+ z − 2
2z
), (6.24a)
B = − αZLd−2z
2−z−∆z
COOe2Z(d+z−22z
)2Z(2∆−d−z2z
)
× ∆(∆ + 2− d− z)
2Z(d+ z − 2∆
2z
)Ψ
(∆
z,d+ z − 2
2z
). (6.24b)
– 41 –
The computation of COO in a holographic Lifshitz theory proceeds along the lines
of Section 3.1.1. Employing (6.6) and (6.16) we find
COO =2∆− d− zz
2∆−d−zz
Z(d+z−2∆2z
)
Z(2∆−d−z2z
). (6.25)
Plugging this result into (6.24), we see complete agreement with (6.14).
7 Sum Rules
One of the main applications of the high frequency asymptotics that we have been
discussing is the derivation of sum rules for dynamical response functions. The most
familiar sum rules are associated with the conductivity σ(ω). Such conductivity sum
rules in near-critical systems were shown recently in [20], and we review the arguments
here, and generalize them to other response functions. Our first goal is to understand
when the following identity holds:
∞∫0
dω Re(σ(ω;T, h)− CJJ · (iω)
d−2z
)= 0, (7.1)
where d is the number of spatial dimensions. This is a highly non-trivial constraint
on the conductivity, which is a scaling function of 2 variables: ω/T and h/T . At
z = 1, (7.1) was derived under certain conditions for CFTs, which we review below
[8, 13, 20, 41]. We will also analyze its validity for Lifshitz theories z 6= 1.
The asymptotic expansion (6.2) allows us to understand precisely when (7.1) will
hold. We need the first subleading term in an asymptotic expansion of σ(ω) to decay
faster than ω−1 as ω → ∞. If this is the case, then we may evaluate the integral of
(7.1) via contour integration, choosing a semicircular contour in the upper half plane
and using analyticity of σ(ω) to show that no poles or branch cuts may be enclosed by
the contour (see e.g. [8, 11, 20]). From (6.2), we conclude that the sum rule is valid
when
d+ z − 2 < ∆ < 2, (7.2)
a result first shown in [20]. The first inequality (the lower bound on ∆) comes from
demanding that the A term decay faster than ω−1; the second inequality (the upper
bound on ∆) comes from demanding the same of the B term. Hence, exactly at the
critical point (h = 0), we find the somewhat weaker bound that ∆ > d− 2 + z, which
agrees with the result for d = 2 CFTs [8], i.e. ∆ > 1.
– 42 –
For general d and z, the analogue of (7.1) for the scalar 2-point function 〈O0O0〉 is
∞∫0
dω Im
(〈O0(ω)O0(−ω)〉
ω− CO0O0
(iω)2∆0−d−z
z
ω
)= 0. (7.3)
where Im(〈O0(ω)O0(−ω)〉) is the usual spectral function. The generalization of (1.3)
to z 6= 1 is
〈O0(ω)O0(−ω)〉 = (iω)2∆0−d−z
z
[CO0O0 +
Ah(iω)1+ d−∆
z
+B〈O〉(iω)
∆z
+ · · ·]. (7.4)
In this paper, we have demonstrated this asymptotic expansion carefully for CFTs
(with z = 1), but we expect that it also holds true for a large class of systems with
z 6= 1 (including the Lifshitz holographic models described above). If 2∆0 > d + z,
then the subtraction in (7.3) is not badly behaved near ω = 0, where we expect that at
any finite T or h, any divergences in 〈O0O0〉 will be resolved. From (7.4), we conclude
that (7.3) holds so long as
2∆0 − d− z < ∆ < 2(d+ z −∆0), (7.5)
using similar logic to the previous paragraph. As with the conductivity, the upper
bound on ∆ is only needed if h 6= 0. We note that (7.2) is a special case of (7.5), when
∆0 = d+z−1, consistent with the fact that in a Lifshitz theory the operator dimension
of Jx is d+ z − 1 (measured in units of inverse length).
We now turn to sum rules for the shear viscosity. These were studied in certain
critical phases (CFTs without a relevant singlet O) in [11–13]. Here we generalize
to quantum critical points with general z, and also to theories detuned by a relevant
operator O. We expect the asymptotic form of the viscosity to be
η(ω) = (iω)dz
[CTT +
Ah(iω)1+ d−∆
z
+B〈O〉(iω)
∆z
+BTP
(iω)1+ dz
+ · · ·]. (7.6)
for generic d and z. As we will shortly see, it is important to keep track of a term
proportional to the pressure P = 〈Txx〉 in the asymptotic expansion. Such a term
can arise due to the stress tensor appearing in the OPE of TxyTxy; for CFTs, we have
already discussed the presence of the stress tensor in the OPE in (4.24). We postulate
that this term will generically occur, as all Lifshitz theories will have a conserved stress
tensor of dimension [Txx] = d+ z, and we confirm that it arises in holographic theories
in Appendix E. A formula for BT 6= 0 has been derived for the conductivity in [8] and
– 43 –
the viscosity in [42], when z = 1; it is related to the coefficients of 〈TTT 〉0 in the CFT
[17]. We may employ (7.5) to find the analogous sum rule
∞∫0
dω Re(η(ω)− CTT (iω)
dz
)= −π
2BTP. (7.7)
The factor of π/2 can be derived by contour integration (e.g. [11, 20]); it is related
to the fact that η(ω) = · · · + BTP/(iω) + · · · has a 1/ω term. Eq. (7.7) is satisfied
whenever
∆ > d+ z, (7.8)
exactly at the critical point, h = 0, but can never be satisfied when h 6= 0 (as the
analogous constraint on ∆ becomes ∆ < 0, which is inconsistent with unitarity).
At finite temperature but zero detuning, h = 0, one can construct a sum rule that
holds in the important case where O is relevant, ∆ < d+ z,
∞∫0
dω Re(η(ω)− CTT (iω)
dz − B〈O〉T (iω)
d−∆z
)= −π
2BTP. (7.9)
This sum rule was introduced for CFTs (z = 1) in [17]. It is more complicated than
the conductivity sum rule (7.1) because one needs to evaluate the 1-point function of
〈O〉T at finite temperature.
Finally, we mention another type of sum rule, called the “dual sum rule”, for the
inverse conductivity 1/σ:∫ ∞0
dω
[Re
(1
σ(ω)
)− 1
CJJ(iω)d−2z
]= 0 (7.10)
which was initially found for CFTs in d = 2 [8, 41]. In that context, it acquires
a physical interpretation by virtue of particle-vortex or S-duality [34, 41, 43]. Here
we generalize it to z 6= 1 and d 6= 2. In order for the sum rule to hold, we need
2 + z− d < ∆ < 2d− 2 which follows from the A,B terms in the large ω expansion. In
addition, we need z > d−2 in order for the subtracted term 1/ω(d−2)/z to be integrable
at small frequencies. Interestingly, in the case of d = 2 (D = 2 + 1) CFTs these
constraints reduce to the ones for the direct sum rule for σ, (7.1).
8 Conclusion
In this paper, we have demonstrated the universal nature of the high frequency response
of conformal quantum critical systems (1.3), both at finite temperature and when
– 44 –
deformed by relevant operators. Holographic methods have demonstrated that, for
certain large-N matrix theories, these results remain correct even when the ground
state is far from a CFT. As a consequence, we find non-perturbative sum rules which
place constraints on the low frequency conductivity, regardless of the ultimate fate of the
ground state. Our results are directly testable in quantum Monte Carlo simulations,
and in experiments in cold atomic gases, where proposals for measuring the optical
conductivity have been made [44].
One interesting generalization of (1.3) will occur when the critical theory is de-
formed not by a spatially homogeneous coupling h, but by a random inhomogeneous
coupling h(x). We expect this randomness to modify the form of the asymptotic
expansion (1.3); such an analysis could be performed in large-N vector models or in
holography. This generalization would be relevant for recent numerical simulations that
yielded the dynamical conductivity across the disorder-tuned superconductor-insulator
transition [9].
It is known that in vector large-N models, interacting Goldstone bosons in a su-
perfluid ground state lead to logarithmic corrections to (1.3) [20]. The likely reason
why this effect is missing in holography is that in these vector large-N models, there
are O(N) Goldstone bosons, whereas holographic superfluids have only one Goldstone
boson. We anticipate that the study of quantum corrections in the bulk, along the
lines of [45], may reveal the breakdown of (1.3) in a holographic superfluid as well.
It would be interesting to understand whether the logarithmic corrections to (1.3) are
controlled universally by thermodynamic properties of the superfluid: the calculation
of [20] suggests that the coefficient of this logarithm is proportional to the superfluid
density.
We have also been able to extend the holographic results for z = 1 QCPs to
z 6= 1. The flavor of the expansion is extremely similar for these theories in holography,
and again we expect that key aspects of this expansion remain true for other Lifshitz
QCPs. It would be interesting to find non-holographic models where this expansion
can be reliably computed.
Acknowledgements
We are grateful to Rob Myers for collaboration in the early stages of this work, and
Tom Faulkner for discussions. We also thank Snir Gazit and Daniel Podolsky for
their collaboration on a closely related project. AL was supported by the Gordon
and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4302. TS was
supported by funding from an NSERC Discovery grant. WWK was supported by a
postdoctoral fellowship and a Discovery Grant from NSERC, by a Canada Research
– 45 –
Chair, and by MURI grant W911NF-14-1-0003 from ARO. WWK further acknowledges
the hospitality of the Aspen Center for Physics, where part of this work was done,
and which is supported by the National Science Foundation through the grant PHY-
1066293. Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development and by the
Province of Ontario through the Ministry of Research & Innovation.
A Two-Point Functions when ∆−D/2 is an Integer
In this appendix, we consider the asymptotics of the correlation function 〈O0(−Ω)O0(Ω)〉when ∆−D/2 is a non-negative integer.
A.1 ∆ = D/2
The most interesting case is ∆ = D/2. The first thing to note is that the momentum
space two-point function of O has logarithms:
〈O(p)O(−p)〉0 = COO logp
µ. (A.1)
These logarithms appear for all ∆ − D/2 = n = 1, 2, . . . as well, multiplied by an
extra factor of p2n. There are two ways to understand the presence of these logarithms.
Firstly, such a logarithm is necessary for the two-point function in position space to be
non-local. Secondly, the emergence of the scale µ breaks conformal invariance, and is
related to conformal anomalies of the dual field theory [28]. We will see below how µ
appears in the asymptotics of two-point functions away from criticality.
A.1.1 Conformal Perturbation Theory
As in the main text, we start by revisiting conformal perturbation theory. The object
that we must study is
〈O0(Ω)O0(−Ω−p)O(p)〉0 ≈ AO0O0O
∞∫0
dx xD2−1Ω2∆0−DK∆0−D
2(Ωx)2K0(px)+O
(p2
Ω2
).
(A.2)
– 46 –
Since the integral is dominated by regions where Ωx ∼ 1, we can approximate this
integral by
〈O0(Ω)O0(−Ω− p)O(p)〉0 ≈ AO0O0OΩ2∆0− 3D2
∞∫0
dy yD2−1K∆0−D
2(y)2×
(− log y − log
p
Ω+ log 2− γe
)= AO0O0OΩ2∆0− 3D
2
[C + Ψ
(D2,∆0 − D
2
)log
Ω
p
]= AO0O0OΩ2∆0− 3D
2 Ψ(D2,∆0 − D
2
) [log
Ω
Cµ− log
p
µ
](A.3)
with C and C complicated constants, and µ an arbitrary scale. The presence of µ in
the answer indicates the running of the couplings, as previously noted.
Our main result (1.3) becomes modified to
〈O0(−Ω)O0(Ω)〉 = Ω2∆0−D(CO0O0 + Ω−
D2
(Ah log
Ω
µ+ B〈O〉
)+ · · ·
). (A.4)
Following the discussion in the main text we find:
A = AO0O0OΨ(D2,∆0 − D
2
), (A.5a)
B = − 1
COOAO0O0OΨ
(D2,∆0 − D
2
)= − ACOO
. (A.5b)
A.1.2 Holography
We now turn to the holographic computation. The first thing we must revisit is the
holographic renormalization prescription for Φ. Following [46], we obtain
Φ(r → 0) =r
D2
LD−1
2
[h log
eγeµr
2− 2〈O〉
]+ · · · , (A.6)
with µ an RG-induced scale and γe the Euler-Mascheroni constant. The specific defi-
nition of µ above will prove useful. We compute COO as in Section 3.1.1. Looking for
solutions to the bulk wave equation of the form Φ(r)e−iΩt, we find
Φ(r) = rD2 K0(Ωr) = r
D2 [− log(Ωr) + log 2− γe + · · · ] . (A.7)
Comparing (A.6) and (A.7) to (A.1) directly gives
COO = −1
2. (A.8)
– 47 –
The computation of AO0O0O then follows very similarly to Section 3.1.2; we find3
AO0O0O =αW
Z(∆0 − D2
)2. (A.9)
The computation of the two-point function 〈O0(Ω)O0(−Ω)〉 then follows very closely
the computation of Section 4.1. Employing the notation introduced there, we find that
we must solve a differential equation analogous to (4.12):
∂2Rχ+
1− 2b
R∂Rχ− χ ≈
αWR2
(h log
eγeµR
2Ω− 2〈O〉
)R
D2
ΩD2
(A.10)
The computation proceeds in an analogous fashion, and we find that the leading cor-
rection to χ is, similarly to (4.18):
χ1(R→ 0) ≈ −Ib(R)Rb
∞∫0
dR0 R0Kb(R0)
Z(b)
αW
ΩD2
RD2−2
0
(h log
eγeµR0
2Ω− 2〈O〉
)
= Ib(R)Rb αW
Z(b)ΩD2
(h log
Ω
Cµ+ 2〈O〉
). (A.11)
Hence, we obtain
〈O0(Ω)O0(−Ω)〉 = Ω2∆0−D
[CO0O0 +
αW
Z(∆0 − D2
)2ΩD2
(h log
Ω
Cµ+ 2〈O〉
)+ · · ·
].
(A.12)
Remarkably, the exotic factor of C defined in (A.3) shows up in the logarithm in (A.12).
As we already defined the scale µ in (A.6) so that the two-point function 〈O(Ω)O(−Ω)〉0took exactly the form (A.1), the appearance of the same C in the logarithms of both
(A.4) and (A.12) is not trivial. Noticing this, and comparing (A.5), (A.8) and (A.9),
we see that (A.12) reduces to (A.4). Hence, the precise agreement of holography to
conformal perturbation theory extends to the special operator dimension ∆ = D/2.
Let us note the important feature that if we re-define µ from our definition in (A.6),
this necessitates a re-definition of 〈O〉. Thus, 〈O〉 hence depends on the choice of µ. It
is crucial that 〈O〉 and µ enter the two-point function in such a way that it is invariant
under a simultaneous shift of µ and 〈O〉. Indeed, comparing (A.6) and (A.12) shows
that this is the case.
3There is a minus sign difference from (3.16), due to the asymptotic behavior of K0(x) ∼ − log x,
instead of + log x.
– 48 –
A.2 ∆ > D/2
The case ∆ > D/2 is much more similar to the case studied in the main text, and so
we give a qualitative treatment, beginning with holography. Let us define
∆ =D
2+ n. (A.13)
The asymptotic behavior of Φ is [28]
Φ(r → 0) =r
D2
LD−1
2
[h
rn(1 + · · ·+ c2nr
2n log(µr))
+ c2nrn〈O〉
]. (A.14)
The coefficients c2n and c2n are dimensionless numbers; their precise values are not
important for this discussion. The crucial thing we notice is that h is the leading order
term, and has no logarithmic corrections. So while there might be further logarithmic
corrections (at higher orders in h), the qualitative form of (4.12), and hence (1.3),
remains correct.
A similar result can be found using conformal perturbation theory. The leading
order correction to the CFT two-point function is proportional to h/ΩD−∆, with no
logarithmic corrections.
B Evaluating CabcdCabcd
In this appendix we derive the equations of motion for hxy(t, r). As in the computations
of 〈O0O0〉 and σ(Ω), we may (to leading order) set the background metric g to that of
pure AdS. For the Weyl correction to the action, it is easiest to first evaluate the action
in terms of hxy. As the Weyl tensor vanishes for pure AdS space, we need only compute
it to first order in h. Up to the action of symmetries, the non-vanishing components of
Cabcd at this order in h are:
Crxty = Crytx = − L2
2r2∂r∂thxy, (B.1a)
Crxry = − L2
2r2
[D − 2
D − 1∂2rhxy −
1
D − 1∂2t hxy
], (B.1b)
Ctxty =L2
2r2
[1
D − 1∂2rhxy −
D − 2
D − 1∂2t hxy+
], (B.1c)
Czxzy =L2
2r2
∂2rhxy + ∂2
t hxyD − 1
. (B.1d)
– 49 –
In the above equations, z stands for a generic spatial direction which is not x or y.
Hence, we find
√gY (Φ)CabcdC
abcd =2
D − 1
(L
r
)D−3
Y (Φ)[2(D − 1)(∂r∂thxy)
2 − 2∂2t hxy∂
2rhxy
+(D − 2)(∂2rhxy)
2 + (D − 2)(∂2t hxy)
2]
(B.2)
The relevant contributions to the equations of motion are
0 =δS
δhxy=r2
L2
[Rxy +
D
r2hxy
]+
δ
δhxy
∫dr√gY (Φ)CabcdC
abcd (B.3)
Using (B.2) and
Rxy +D
r2hxy =
1
2
[−∂2
rhxy +D − 1
r∂rhxy + r2Ω2hxy
](B.4)
we obtain (4.38).
C Computing ATTO
In this appendix, we assume that all momenta p1,2,3 point in the t direction. Start-
ing with (B.2), and slightly rearranging, the bulk action which we must evaluate to
determine ATTO is given by
Sbulk =
∫dD+1x
√gg−1g−1 4αZL
D+32
κ2Φ(p3) [∂µ∂νhxy(p1)∂µ∂νhxy(p2)
−∂µ∂µhxy(p1)∂ν∂νhxy(p2)
D − 1
]. (C.1)
In this appendix, we will denote with g−1 = r2/L2 a generic diagonal element of the AdS
metric (3.3);√g = (L/r)D+1 is unchanged. We will also use an Einstein summation
convention on lowered Greek indices, with no factors of the AdS metric included.
We now perform a series of integration by parts manipulations, with the end goal
to remove all derivatives from hxy. We begin by analyzing the second term in more
detail. Integration by parts a few times leads to∫dr√gg−1g−1Φ(p3)∂µ∂µhxy(p1)∂ν∂νhxy(p2)
=
∫dr[√
gg−1g−1Φ(p3)∂ν∂µhxy(p1)∂µ∂νhxy(p2)
+ ∂µ∂ν(√
gg−1g−1Φ(p3))∂µhxy(p1)∂νhxy(p2)
− ∂µ∂µ(√
gg−1g−1Φ(p3))∂νhxy(p1)∂νhxy(p2)
]. (C.2)
– 50 –
After more integration by parts, the second term on the right hand side above can be
replaced with ∫dr∂µ∂ν
(√gg−1g−1Φ(p3)
)∂µhxy(p1)∂νhxy(p2)
=
∫dr[1
2∂µ∂µ
(√gg−1g−1Φ(p3)
)∂νhxy(p1)∂νhxy(p2)
+√gg−1g−1Φ(p3)
(∂νhxy(p1)∂ν∂µ∂µhxy(p2)
+ ∂ν∂νhxy(p1)∂µ∂µhxy(p2))]. (C.3)
We are finally ready to employ the Ω =∞ equations of motion,
∂2r −
D − 1
r∂rhxy − r2Ω2hxy = 0, (C.4)
to simplify the second term above:∫dr√gg−1g−1Φ(p3)∂νhxy(p1)∂ν∂µ∂µhxy(p2) (C.5)
=
∫dr√gg−1g−1Φ(p3)∂νhxy(p1)∂ν
(D − 1
r∂rhxy(p2)
)=
∫dr√gg−1g−1Φ(p3)
[D − 1
2r∂r (∂νhxy(p1)∂νhxy(p2))− ∂µ∂µhxy(p1)∂ν∂νhxy(p2)
D − 1
].
(C.6)
In the second line above, we have re-used the equation of motion (C.4) to replace
∂rhxy∂rhxy. Combining (C.2), (C.3) and (C.6), we find
S ≡∫
dr√gg−1g−1Φ(p3)
[∂ν∂µhxy(p1)∂µ∂νhxy(p2)− ∂µ∂µhxy(p1)∂ν∂νhxy(p2)
D − 1
]=
1
2
∫dr∂νhxy(p1)∂νhxy(p2)
[∂µ∂µ
(√gg−1g−1Φ(p3)
)+ ∂r
(D − 1
r
√gg−1g−1Φ(p3)
)].
(C.7)
From (C.1), S is proportional to Sbulk. Straightforward integration by parts, along with
the equation of motion (C.4), gives us∫dr√gg−1∂νhxy(p1)∂νhxy(p2)Y =
∫dr
2hxy(p1)hxy(p2)∂µ
(√gg−1∂µY
). (C.8)
Upon using the equations of motion for Φ, we find
∂µ∂µ(√
gg−1g−1Φ(p3))
+ ∂r
(D − 1
r
√gg−1g−1Φ(p3)
)=√gg−1Y (C.9)
– 51 –
where
Y ≡ ∆(∆−D)− 2(D − 2)
2L2Φ(p3) +
2r
L2∂rΦ(p3)
=(∆ + 2)(∆ + 2−D)
2L2Φ(p3) +
2rD−∆
Ld2Z(∆− D
2)K∆−D
2+1(p3r) (C.10)
Upon using the identities (3.24) we find
∂µ(√
gg−1∂µY)
=∆(∆−D)(∆ + 2)(∆ + 2−D)
2L4
√gΦ(p3)
+2(∆ + 2)(D −∆− 2)
L4
√grD−∆K∆−D
2+1(p3r)
Ld/2Z(∆− D2
)+
4
L4
√grD−∆K∆−D
2+2(p3r)
Ld/2Z(∆− D2
).
(C.11)
Combining (C.1) and (C.11) with
〈Txy(p1)Txy(p2)O(p3)〉0 = − δ3Sbulk
δhxy(p1)δhxy(p2)δΦ(p3)(C.12)
we recover (2.38) with ATTO given by (3.29).
D Generalization of Holographic Asymptotic Integrals
In this appendix, we discuss some of the possibilities which we neglected in the main
text in our derivation of (4.36). The notation of this appendix follows the discussion
around (4.18).
D.1 a+ 2 < 2b
As a > 0 (0 < ∆0 < D), we may safely assume that b > 0. What happens in this case
is that the integral in (4.18) is divergent. As a consequence, we must explicitly write
χ = cRbKb(R)
R∫0
dR0 R1+a0 Kb(R0)Ib(R0) + cRbIb(R)
∞∫R
dR0 R1+a0 Kb(R0)2. (D.1)
We now analyze the integrands as R,R0 → 0. Denoting p(x) as an arbitrary regular
Taylor series in x (no divergences or non-analytic contributions), the first integral in
(D.1) is
RbKb(R)
R∫0
dR0 R1+a0 Kb(R0)Ib(R0) = (p(R)+R2bp(R))
R∫0
dR0R1+a0 (p(R0)+R2b
0 p(R0))
= p(R)R2+a + p(R)R2+a+2b + p(R)R2+a+4b. (D.2)
– 52 –
Hence, this integral cannot contribute to the O(R2b) term. The second integral is
divergent as R→ 0 and can be analytically evaluated to:
∞∫R
dR0 R1+a0 Kb(R0)2 = Ψ(a+ 2; b) +R2+a−2bp(R) +R2+ap(R) +R2+a+2bp(R), (D.3)
with p(R)s above given by specific hypergeometric functions. Hence, we see that the
only term which will scale as R2b is proportional to Ψ(a + 2; b). This justifies the
“analytic continuation” of a+ 2− 2b through 0 using holography.
D.2 b < 0
When b < 0, the dominant term in ψ(R) as R→ 0 is the response, and the subleading
term is the source. Unitarity in a conformal field theory requires that for any non-trivial
scalar operator [47]
∆,∆0 >D
2− 1, (D.4)
which implies that b > −1. Now, when b < 0, xbKb(x) ∼ x2b as x→ 0. (4.18) converges
so long as a+ 1 + 2b > −1, which becomes equivalent to a > 0. Since we have assumed
that the operator O is relevant, this is indeed the case. Thus, the computation of the
response presented in the main text remains correct.
E Viscosity in a Holographic Lifshitz Model
In this appendix we argue that the BT term in (7.6) arises in a generic holographic
model. For simplicity, we focus on a background geometry described by the “Lifshitz
black brane” (in Euclidean time):
ds2 =L2
r2
[dr2
f(r)+f(r)
r2z−2dt2e + dx2
], f(r) = 1−
(r
r+
)d+z
. (E.1)
The constant r+ is related to the temperature, and in particular the thermodynamic
pressure obeys
P ∝ r−d−z+ . (E.2)
More generally, we expect that the pressure is related to the O(rd+z) contributions to
f(r), though we note that the holographic renormalization for Tµν can require some
care, especially in Lifshitz theories [48]. With this particular choice for f(r), we find
that for a rather general family of x-independent matter backgrounds that support a
– 53 –
Lifshitz geometry, the equations of motion for the metric perturbation hxy, defined in
(4.37), are independent of the details of the bulk matter:
0 = rd+z+1f∂r
(f
rd+z−1∂rhxy
)− r2zΩ2hxy. (E.3)
In terms of the coordinate R defined in (6.8) we find
0 = Rdz f
((zR
Ω
) 1z
)∂R
(1
Rdz
f
((zR
Ω
) 1z
)∂Rhxy
)− hxy. (E.4)
When Ω = ∞, we can approximate f = 1; the resulting equation is regular in R and
we may solve it with suitable boundary conditions with a modified Bessel function, as
before. Hence about Ω =∞ we look for a solution of the form hxy = h(0)xy + h
(1)xy where
Rdz ∂R
(R−
dz ∂Rh
(1)xy
)− h(1)
xy =1
rd+z+
(zR
Ω
)1+ dz
h(0)xy +R
dz ∂R
(1
rd+z+ R
dz
(zR
Ω
)1+ dz
∂Rh(0)xy
).
(E.5)
We conclude by linearity and (E.2) that h(1)xy ∝ PΩ−1− d
z . As h(1)xy is sourced by a
complicated function, we do not expect the O(R1+ d
z
)coefficient of h
(1)xy , responsible
for the corrections to η(iΩ), will be non-zero. Hence, the coefficient BT in (7.6) must
be generically non-zero.
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